Diophantine approximation as Cosmic Censor for Kerr-AdS black holes

The purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant $\Lambda<0$ and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations $\psi$ of Kerr-AdS solving $\Box_g\psi-\frac{2}{3}\Lambda \psi=0$ with reflecting boundary conditions at infinity. Understanding the behavior of $\psi$ at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass $\mathfrak m = M \sqrt{-\Lambda}$ and angular momentum $\mathfrak a = a \sqrt{-\Lambda}$ satisfy a certain non-Diophantine condition, then perturbations $\psi$ arising from generic smooth initial data blow up at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner-Nordstr\"om-AdS (arxiv:1812.06142) as well as to previous work on the analogous problem for $\Lambda \geq 0$. As a result of the non-Diophantine condition, the set of parameters $\mathfrak m, \mathfrak a$ for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking-Reall bound. On the other hand, we conjecture that for a set of parameters $\mathfrak m, \mathfrak a $ which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon. This suggests that the validity of the $C^0$-formulation of Strong Cosmic Censorship for $\Lambda<0$ may change in a spectacular way according to the notion of genericity imposed.


Introduction
The Kerr-Anti-de Sitter (Kerr-AdS) black hole spacetimes (M, g) constitute a 2-parameter family of solutions to the celebrated Einstein equations Ric µν (g) − 1 2 Rg µν + Λg µν = 8πT µν (1.1) in vacuum (T µν = 0) and with negative cosmological constant Λ < 0. The family (see already (2.14) for the metric) is parameterized by the black hole mass M > 0, and specific angular momentum a = 0. The Kerr-AdS black holes posses a smooth Cauchy horizon beyond which the spacetime has infinitely many smooth extensions-thus violating determinism. Regular Cauchy horizons are thought, however, to be generically unstable, which is the content of the Strong Cosmic Censorship conjecture due to Roger Penrose [88]. Its strongest formulation, the C 0formulation [8] (see already Conjecture 1), states that for generic initial data for (1.1), the metric cannot be continuously extended beyond a Cauchy horizon, in this sense saving determinism within classical general relativity. Unfortunately, for Λ = 0 and Λ > 0, this formulation was disproved by Dafermos-Luk [18]. However, a weaker formulation put forward by Christodoulou is still expected to be true (see already Conjecture 2). Refer to Section 1.1 for a more detailed discussion. For Λ < 0, the question of the validity of the C 0 -formulation of Strong Cosmic Censorship has until today remained open. Motivated by the above, we study linear scalar perturbations ψ of subextremal Kerr-AdS black holes solving the conformal scalar wave equation which arise from smooth and compactly supported initial data posed on a spacelike hypersurface and which satisfy reflecting boundary conditions at infinity. We further assume that the black hole parameters satisfy the Hawking-Reall bound [45], see already (2.8). One can view (1.2) as a linear scalar analog of (1.1), and so the linear scalar analog of the C 0 -formulation of Strong Cosmic Censorship is the statement that for generic black hole parameters, linear scalar perturbations ψ, arising from generic initial data for (1.2), fail to be continuous at the Cauchy horizon (see already Conjecture 3).
Our main result Theorem 1 shows that if the dimensionless Kerr-AdS parameters mass m = M √ −Λ and angular momentum a = a √ −Λ satisfy a certain non-Diophantine condition, then linear scalar perturbations ψ solving (1.2) and arising from generic initial data blow up |ψ| → +∞ (1.3) at the Cauchy horizon. We show that the set of such parameters is Baire-generic (but Lebesgueexceptional ). Hence, our main result provides an-unexpected-positive resolution of the linear scalar analog of the C 0 -formulation of the Strong Cosmic Censorship conjecture for Λ < 0, provided that the genericity of the set of parameters is taken in the Baire-generic sense.
Theorem 1 is in sharp contrast to the result on Reissner-Nordström-AdS black holes [61] and to previous work on Strong Cosmic Censorship for Λ ≥ 0-in both cases such perturbations ψ were shown to remain bounded and to extend continuously across the Cauchy horizon.
The instability result (1.3) of Theorem 1 is not associated to superradiance (since the parameters satisfy the Hawking-Reall bound) and, more surprisingly, is also not a consequence of the well-known blue-shift instability [87] at the Cauchy horizon. Instead, Theorem 1 is a manifestation of the occurrence of small divisors originating from a new resonance phenomenon between, on the one hand, high frequencies associated to stable trapping on the exterior [54,56] and, on the other hand, the poles of the interior scattering operator which are characteristic frequencies with respect to the Killing generator of the Cauchy horizon [62]. For this, it is fundamental that Kerr-AdS is rotating, as it is only in this case that stably trapped high frequency waves can, at the same time, be characteristic frequency waves with respect to the Killing generator of the Cauchy horizon. If now m, a satisfy the non-Diophantine condition, then the resonance will be sufficiently strong (and the occurring divisors will be sufficiently small) so as to cause the instability (1.3).
Thus, in the case Λ < 0, surprisingly, Diophantine approximation may turn out to be the elusive "Cosmic Censor" which Penrose was searching for in order to protect determinism in general relativity [88].
The story, however, has yet another level of complexity. We also conjecture that, if the dimensionless black hole parameters m = M √ −Λ and a = a √ −Λ satisfy a Diophantine condition, then linear scalar perturbations ψ remain bounded |ψ| ≤ C at the Cauchy horizon. This would then hold for Lebesgue-generic but Baire-exceptional black hole parameters. If true, this would provide a negative resolution of the linear scalar analog of the C 0 -formulation of Strong Cosmic Censorship provided that genericity of the parameters is now taken in the Lebesgue-generic sense.
Returning to the fully nonlinear C 0 -formulation of Strong Cosmic Censorship, the black hole parameters are themselves dynamic in evolution under (1.1). Thus, the above competing notions of genericity for the parameters may now be reflected in different formulations of the genericity assumption imposed on initial data in the statement of the conjecture. This could mean that the validity of Strong Cosmic Censorship is not only sensitive to the regularity of the extension but may also become highly sensitive to the precise notion of genericity imposed on the initial data.

Outline of the introduction
We begin in Section 1.1 with a presentation of the C 0 -formulation (Conjecture 1) and Christodoulou's reformulation (Conjecture 2) of the Strong Cosmic Censorship conjecture. We also introduce their respective linear scalar analogs Conjecture 3 and Conjecture 4 and review the relevant previous work and difficulties for Λ ≥ 0 and Λ < 0. Then, turning to the Kerr-AdS case (Λ < 0), we will first outline in Section 1.2 the behavior of linear scalar perturbations on the black hole exterior before we focus on the interior in Section 1.3, see Fig. 1. In Section 1.4 we put both insights together and we will see, at least on a heuristic level, how small divisors and Diophantine approximation arise. This will lead to a new expectation that transcends Conjecture 3 and Conjecture 4 and which we formulate in Section 1.5 as Conjecture 5 and Conjecture 6. In Section 1.6 we state our main result Theorem 1, which resolves Conjecture 5 in the affirmative. Then, in Section 1.7 we give an outlook on Conjecture 6. In Section 1.8 we describe how we turn our heuristics of Section 1.4 into a proof of Theorem 1. Finally, we give a brief outline of the paper in Section 1.9.

Strong Cosmic Censorship: Conjectures 1-4
Recall from our previous discussion that our main motivation for studying linear perturbations on black hole interiors is to shed light on one of the most fundamental problems in general relativity: the existence of smooth Cauchy horizons.
In general, a Cauchy horizon CH defines the boundary beyond which initial data on a spacelike hypersurface (together with boundary conditions at infinity in the asymptotically AdS case) no longer uniquely determine the spacetime as a solution of the Einstein equations (1.1). The Kerr(de Sitter or -Anti-de Sitter) black holes share the property that they indeed posses a smooth Cauchy horizon CH in their interiors. In particular, these spacetimes admit infinitely many smooth extensions beyond their Cauchy horizons solving (1.1), and in this sense violating determinism and the predictability of the theory. From a PDE point of view, this corresponds to a lack of global uniqueness for (1.1). However, the existence of regular Cauchy horizons is conjectured to be an artifact of the high degree of symmetry in those explicit spacetimes and generically it is expected that some sort of singularity ought to form at or before a Cauchy horizon. The original mechanism which was invoked to support this expectation is a blue-shift instability associated to Cauchy horizons [87]. The emergence of such a singularity at or before a Cauchy horizon is paradoxically "good" because-if sufficiently strong-it can be argued that this restores determinism, as the fate of any observer approaching the singularity, though bleak, is uniquely determined. Making this precise gives rise to various formulations of what is known as the Strong Cosmic Censorship (SCC) conjecture [88,10].
We begin with the C 0 -formulation of the SCC conjecture which can be seen as the strongest, most desirable, inextendibility statement in this context.
Conjecture 1 (C 0 -formulation of Strong Cosmic Censorship). For generic compact, asymptotically flat or asymptotically Anti-de Sitter vacuum initial data, the maximal Cauchy development of (1.1) is inextendible as a Lorentzian manifold with C 0 metric. This formulation is related to the statement that observers are torn apart by infinite tidal deformations before they have the chance to cross a Cauchy horizon [84,18].
Surprisingly, the C 0 -formulation (Conjecture 1) was recently proved to be false for both cases Λ = 0 and Λ > 0 [18] (see discussion later). The reason is that it turns out that the blue-shift instability is not sufficiently strong to destroy the metric itself, only derivatives of the metric. However, the following weaker, yet still well-motivated, formulation introduced by Christodoulou in [10] is still expected to hold true (though for the Λ > 0 case see the discussion later).

Conjecture 2 (Christodoulou's reformulation of Strong Cosmic Censorship).
For generic compact, asymptotically flat or asymptotically Anti-de Sitter vacuum initial data, the maximal Cauchy development of (1.1) is inextendible as a Lorentzian manifold with C 0 metric and locally square integrable Christoffel symbols.
Unlike the C 0 -formulation in Conjecture 1, the statement of Conjecture 2 does not guarantee the complete destruction of observers approaching Cauchy horizons. However, it restores determinism in the sense that even just weak solutions must break down at Cauchy horizons. Nonetheless, one may remain uneasy as to whether the standard notion of weak solution to (1.1) is finally the correct one [86,71,73]. In this sense it is a pity that Conjecture 1 turned out to be false in the Λ ≥ 0 cases, as it would have provided a much more definitive resolution of the spirit of the Strong Cosmic Censorship conjecture. Hence, it is of interest to know whether the situation is better in the Λ < 0 case! Linear scalar analog of the Strong Cosmic Censorship conjecture The aforementioned formulations of SCC have linear scalar analogs on the level of (1.2). Indeed, under the identification ψ ∼ g, the linear scalar wave equation (1.2) can be seen as a naive linearization of the Einstein equations (1.1) after neglecting the nonlinearities and the tensorial structure. Moreover, many phenomena and difficulties for the full Einstein equations (1.1) are already present at the level of (1.2).
The linear scalar analog of Conjecture 1 in a neighborhood of Kerr and Kerr-(Anti-)de Sitter corresponds to the statement that for generic black hole parameters, linear scalar perturbations ψ arising from generic data on a spacelike hypersurface solving (1.2) blow up in amplitude at the Cauchy horizon.
The reformulation due to Christodoulou (Conjecture 2) finds its linear scalar analog in the H 1 loc blow up of ψ at the Cauchy horizon in view of the identification ∂ψ ∼ Γ.
The word generic appears twice in the above formulations, both in the context of the parameters and in the context of the perturbation. This is because in the fully nonlinear Conjecture 1 and Conjecture 2, the background parameters are themselves dynamic in evolution under (1.1) and thus both would be encompassed in the genericity of the initial data.
Genericity of the black hole parameters. As we will show in the present paper, for the Kerr-AdS case, the validity of Conjecture 3 and Conjecture 4 will depend in a crucial way on the notion of genericity (Baire-generic or Lebesgue-generic) imposed on the parameters. This will eventually lead us to refine the above statements of Conjecture 3 and Conjecture 4 (see already Section 1.5).
Genericity of the initial data. We will assume that the initial data lie in the class of smooth functions of compact support. Regarding genericity within that class, note that just finding one single solution for which the blow-up statement is true already yields a natural notion of genericity. Indeed, since (1.2) is linear, it would then follow that data for which the arising solution does not blow up satisfy a co-dimension 1 property (see already Remark 1.1) and thus, would be exceptional. It is this notion of genericity of the initial data which we will consider later in Section 1.5. Note that we will also consider a more refined notion of genericity of initial data in Remark 1.5.
Before we bring our discussion of SCC to asymptotically AdS black holes (Λ < 0), we will first review the state of the art of the SCC conjecture for the cases Λ = 0 and Λ > 0.
SCC for Λ = 0 and Λ > 0 Linear level. The definitive negative resolution of the fully nonlinear Conjecture 1 in [18] for both Λ = 0 and Λ > 0 was preceded by the negative resolution of the linear Conjecture 3 in [34,35,46] for Λ = 0 and in [48,12] for Λ > 0. It was shown that solutions of (1.2) arising from regular and localized data on a spacelike hypersurface remain continuous and uniformly bounded |ψ| ≤ C at the Cauchy horizon for all subextremal Kerr black hole interiors (Λ = 0), and very slowly rotating subextremal Kerr-dS black hole interiors-hence disproving Conjecture 3 for Λ = 0 and Λ > 0. (For the extremal case see [38,39] and for the Schwarzschild case see [32].) The key ingredient in showing boundedness at the Cauchy horizon is a sufficiently fast decay (polynomial with rate v −p with p > 1 for Λ = 0 and exponential for Λ > 0) of linear scalar perturbations along the event horizon. Using suitable energy estimates associated to the red-shift vector field introduced in [19] and the vector field S = |u| p ∂ u + |v| p ∂ v , this decay is then propagated into the black hole all the way up to the Cauchy horizon CH, where it is sufficient to conclude uniform boundedness. We remark already that this method manifestly fails for asymptotically AdS black holes, where linear scalar perturbations decay merely at a logarithmic rate along the event horizon [54,56].
While Conjecture 3 is false for Λ = 0, as remarked above, at least the weaker formulation Conjecture 4 holds true: It was proved that the (non-degenerate) local energy at the Cauchy horizon blows up, ψ H 1 loc = +∞, for a generic set of solutions ψ on Reissner-Nordström [68] and Kerr [20] black holes in the full subextremal range of parameters. A similar blow-up behavior was obtained for Kerr in [72] assuming lower bounds (which were shown later in [47] to indeed hold generically) on the energy decay rate of a solution along the event horizon. These results thus also support the validity of the fully nonlinear Conjecture 2 for Λ = 0.
On the other hand, in the Λ > 0 case, the exponential convergence of perturbations along the event horizon of a Kerr-de Sitter black hole is in direct competition with the exponential blue-shift instability near the Cauchy horizon. Thus, the question of the validity of Conjecture 4 becomes even more subtle for Λ > 0 and has received lots of attention in the recent literature. We refer to the conjecture in [16], the survey article [91], the recent results [21,24,23,22,12] and the works [51,50] taking also quantum effects into account.
Another related result, which will turn out to be important for the paper at hand, is proved in work of the author and Shlapentokh-Rothman [62]: The main theorem establishes a finite energy scattering theory for solutions of (1.2) on the interior of Reissner-Nordström. In this scattering theory, a linear isomorphism between the degenerate energy spaces (associated to the Killing field T ) corresponding to the event and Cauchy horizon is established. The problem reduces to showing uniform bounds for the transmission and reflection coefficients T(ω, ) and R(ω, ) for fixed frequency solutions. Formally, for an incoming wave at the right event horizon H R , the transmission and reflection coefficients correspond to the amount of T -energy scattered to the left and right Cauchy horizon CH L and CH R , respectively. Indeed, the theory also carries over to non-zero cosmological constant Λ = 0 except for the characteristic frequency (ω = 0) associated to T , thought of now as the generator of the Cauchy horizon. (Note that these results are compatible with the blow-up of the local energy at the Cauchy horizon [68] because of the degeneracy of the T -energy.) These insights will turn out to be important for the interior analysis of the present paper, see already Section 1.3.
Nonlinear level. Turning to the nonlinear problem of (1.1), Dafermos-Luk proved the full nonlinear C 0 -stability of the Kerr Cauchy horizon in [18]. Their work definitively disproves Conjecture 1 for Λ = 0 (subject only to the completion of a proof of the nonlinear stability of the Kerr exterior). Mutatis mutandis, their proof of C 0 -stability also applies to Kerr-de Sitter Cauchy horizons, where the exterior has been shown to be nonlinearly stable in the very slowly rotating case [49]. This unconditionally disproves Conjecture 1 for Λ > 0.
Nonlinear inextendibility results at Cauchy horizons have been proved only in spherical symmetry: For the Einstein-Maxwell-scalar field system, the Cauchy horizon is shown to be C 2 unstable [15,69,70] for a generic set of spherically symmetric initial data. See also the pioneering work in [89,84] and the more general results on the Einstein-Maxwell-charged scalar field system in [93,94,95]. This proves the C 2 -formulation of SCC, and by very recent work [92], the C 0,1formulation (but not yet Conjecture 2) in spherical symmetry. For work in the Λ > 0 case see [13,14]. The question of any type of nonlinear instability of the Cauchy horizon without symmetry assumptions and the validity of Conjecture 2 (even restricted to a neighborhood of Kerr) have yet to be understood.
SCC for asymptotically AdS spacetimes Λ < 0 As we shall see in the present paper, the situation for asymptotically AdS black holes with Λ < 0 will turn out to be radically different. First, in view of the lack of global hyperbolicity of asymptotically AdS spacetimes, one needs to specify additional boundary conditions at infinity (at I) to guarantee well-posedness of (1.1) and (1.2), see [37,33,52,96,42]. The most natural in this context are reflecting (Dirichlet) boundary conditions [37]. In what follows we will assume such Dirichlet boundary conditions. (Refer to Section 1.5 and Section 1.6 later for remarks on more general boundary conditions.) We first discuss linear scalar perturbations solving (1.2) arising from data posed on a spacelike hypersurface on asymptotically AdS black holes. In contrast to Λ ≥ 0, where linear scalar perturbations ψ decay at a polynomial (Λ = 0) and exponential (Λ > 0) rate, linear scalar perturbations ψ of Kerr-AdS (and Reissner-Nordström-AdS) decay merely at a logarithmic rate on the exterior as proved in [54,56]. 1 The origin of this slow decay is a stable trapping phenomenon of high-frequency waves traveling along stably trapped null geodesics which repeatedly bounce off null infinity I. (Contrast this with the work [25,85] in 2+1 dimensions.) For 5D asymptotically flat black holes, a similar log-decay result was shown in [3], which also relies on the existence of stably trapped null geodesics.
With the logarithmic decay on the exterior in hand, we first recall from the discussion above that in the Λ ≥ 0 cases Conjecture 3 is false (and, in fact, so is the fully nonlinear Conjecture 1), yet at least in the Λ = 0 case Conjecture 4 is true (and, hopefully, Conjecture 2 as well). Indeed, our methods in principle also show Conjecture 4 for Λ < 0. However, in view of the slower decay in the case Λ < 0, one might suspect a stronger instability at the Cauchy horizon in this case. This raises the attractive possibility that Conjecture 1 and Conjecture 3 might actually be true for Λ < 0, which would give a more definitive resolution to the issue of Strong Cosmic Censorship than the weaker Conjecture 2 and Conjecture 4.
For the Reissner-Nordström-AdS spacetime, which is often considered as a toy model of Kerr-AdS, this question was first taken up in [61]. For that case, however, the hopes expressed in the above paragraph were not fulfilled! It was shown in [61] that, despite the slow decay on the exterior, all linear scalar perturbations ψ on Reissner-Nordström-AdS (in the full subextremal range) remain uniformly bounded, |ψ| ≤ C, on the interior and extend continuously across the Cauchy horizon. Thus, the Reissner-Nordström analog of Conjecture 3 is false. To understand the additional phenomenon which was exploited to prove boundedness, let us decompose a linear scalar perturbation ψ into frequencies ω, m, associated to the separation of variables. On the exterior, it is the high frequency part (i.e. |ω|, |m|, large) of ψ which is exposed to stable trapping and decays slowly, whereas the low frequency part (|ω|, |m|, small) decays superpolynomially. In the interior, however, the main obstruction to boundedness is the interior scattering pole which is located at the characteristic frequency ω = 0 with respect to T , now thought of as the Killing generator of the Cauchy horizon. (Refer also to the discussion in [62,Section 3.6].) Thus, for Reissner-Nordström-AdS, the slowly decaying part of ψ is decoupled in frequency space from the part susceptible to the interior scattering pole at ω = 0. (See already Fig. 6.) The above result on Reissner-Nordström-AdS may suggest that, just as in the cases of Λ ≥ 0, Conjecture 3 is false for Λ < 0, albeit for different reasons.
The present paper on Kerr-AdS, however, provides an unexpected positive resolution of Conjecture 3 for Λ < 0. We show in Theorem 1 that there exists a set P Blow-up of dimensionless Kerr-AdS parameters m := M √ −Λ and a := a √ −Λ which is Baire-generic but Lebesgue-exceptional, such that on all Kerr-AdS black hole whose parameters lie in P Blow-up , generic linear scalar perturbations ψ blow up |ψ| → +∞ at the Cauchy horizon. Thus, our main result Theorem 1 shows that Conjecture 3 is true if Baire-genericity is imposed on the Kerr-AdS parameters.
This set of parameters is defined through a non-Diophantine condition. This condition arises from small divisors originating from a resonance phenomenon between, on the one hand, specific high frequencies associated to stable trapping on the exterior and, on the other hand, poles of the interior scattering operator which are characteristic frequencies with respect to the Killing generator of the Cauchy horizon. This resonance phenomenon is possible because the characteristic frequencies of the Cauchy horizon are now the frequencies where 1 Recall that we restrict our attention to Kerr black holes below the Hawking-Reall bound [45] as otherwise growing modes are shown to exist [27].
is the frequency at which the Cauchy horizon rotates. In contrast to Reissner-Nordström, (1.6) can now be satisfied for frequencies |ω|, |m|, which are large. It is not all high frequencies, however, but only specific high frequencies, so-called quasimode (on the real axis) or quasinormal mode (in the complex plane) frequencies (ω n , m n , n ) n∈N which are responsible for the slow decay on the exterior. (See already Section 1.2.) This resonance phenomenon will lead to small divisors of the form 1 ωn−ω−mn . Now, if the specific quasinormal mode frequencies approximate the characteristic frequencies ω = ω − m sufficiently well, i.e. if |ω n − ω − m n | is sufficiently small for infinitely many n ∈ N, then we will show that generic linear scalar perturbations ψ of Kerr-AdS blow-up |ψ| → ∞ at the Cauchy horizon. This naturally leads to a non-Diophantine condition on the black hole parameters m, a which, as we will show, holds true for a set of parameters m, a which is Bairegeneric but Lebesgue-exceptional.
The above result is not the last word on Conjecture 3 on Kerr-AdS black holes. We also complement our main result with the conjecture that if the parameters m, a satisfy a complementary Diophantine condition, then the resonance phenomenon outlined above is "weak" and linear scalar perturbations ψ remain bounded |ψ| ≤ C at the Cauchy horizon. This would then hold for black hole parameters which lie in a set P Bounded which is Baire-exceptional but Lebesguegeneric. Thus, we expect Conjecture 3 to be false if Lebesgue-genericity is imposed on the Kerr-AdS parameters.
Since the parameters are dynamic in the full nonlinear (1.1), this suggests that for Λ < 0 the validity of the C 0 -formulation of Strong Cosmic Censorship (Conjecture 1) may change in a spectacular way according to the notion of genericity imposed.
Instability of asymptotically AdS spacetimes? If we accept to interpret the above results as supporting Conjecture 1, they leave determinism in better shape for Λ < 0 compared to the Λ ≥ 0 cases. However, turning to the fully nonlinear dynamics governed by (1.1), there is yet another scenario which could happen. While Minkowski space (Λ = 0) and de Sitter space (Λ > 0) have been proved to be nonlinearly stable [36,11], Anti-de Sitter space (Λ < 0) is expected to be nonlinearly unstable with Dirichlet conditions imposed at infinity. This was recently proved by Moschidis [75,74,77,76] for appropriate matter models. See also the original conjecture in [17] and the numerical results in [5]. Similarly, for Kerr-AdS (or Reissner-Nordström-AdS), the slow logarithmic decay on the linear level proved in [56] could in fact give rise to nonlinear instabilities in the exterior. (Note that in contrast, nonlinear stability for spherically symmetric perturbations of Schwarzschild-AdS was shown for Einstein-Klein-Gordon systems [55].) If indeed the exterior of Kerr-AdS was nonlinearly unstable, the linear analysis on the level of (1.2) could not serve as a model for (1.1) and the question of the validity of Strong Cosmic Censorship would be thrown even more open! 1.2 Exterior: log-decay, quasi(normal) modes and semi-classical analysis We recall the result of Holzegel-Smulevici [54,56] that linear scalar perturbations ψ solving (1.2) decay at a sharp inverse logarithmic rate on the Kerr-AdS exterior. (For smooth initial data, the decay in (1.8) can be slightly improved to |ψ| ≤ Cn log n (t) for n ∈ N.) The reason for the slow decay is the stable trapping phenomenon near infinity discussed earlier. One manifestation of this phenomenon is the existence of so-called quasimodes and quasinormal modes which are "converging exponentially fast" to the real axis. Note already that in the proof of Theorem 1 we will work with quasimode frequencies but we will not make use of a quasinormal mode construction or decomposition. However, quasinormal modes provide perhaps the simplest route to obtain some intuition-paired with the interior analysis in Section 1.3-for how the relation to Diophantine approximation arises. Our discussion of quasi(normal) modes starts with the property that (1.2) is formally separable [7]. for a rescaled radial variable r * ∈ (−∞, π 2 l) with r * (r = r + ) = −∞, r * (r = +∞) = π 2 l. The radial o.d.e (1.10) couples to the angular o.d.e. through the potentialṼ which depends on the eigenvalues λ m (aω) of the angular o.d.e. P (aω)S m (aω, cos θ) = λ m (aω)S m (aω, cos θ), (1.11) where P (aω) is a self-adjoint Sturm-Liouville operator. The radial o.d.e. (1.10) is equipped with suitable boundary conditions at r * = −∞ and r * = π 2 which stem from imposing regularity for ψ at the event horizon and Dirichlet boundary conditions at infinity. This leads to the concept of a mode solution ψ of (1.2) defined to be of the form (1.9) such that u solves (1.10) and S m solves (1.11) with the appropriate boundary conditions imposed. If such a solution ψ were to exist for ω ∈ R, this would correspond to a time-periodic solution. Such solutions are however incompatible with the fact that all admissible solutions decay. Nevertheless, there exist "almost solutions" which are time-periodic. This leads us to the concept of Quasimodes. In [56] it was shown that there exists a set of real frequencies (ω n , m n = 0, n ) n∈N such that the corresponding functions ψ n "almost" solve (1.2) in the sense that they satisfy g ψ n + 2 3 Λψ n = F n with |F n | exp(−n). These almost-solutions are called quasimodes and their existence actually implies that the logarithmic decay of [54] is sharp as shown in [56]. These quasimode frequencies are equivalently characterized by the condition that the Wronskian W[u H + , u ∞ ] of solutions u H + , u ∞ of (1.10) adapted to the boundary conditions satisfies

Separation of Variables. With the fixed-frequency ansatz
(1.12) The reason why there exist such quasimodes is that in the high frequency limit, the potential in (1.10) admits a region of stable trapping, see already Fig. 3. Alternatively and intimately related to the above, the existence of quasimodes can be seen as a consequence of the existence of stably trapped null geodesics on the exterior of asymptotically AdS black holes.
Quasinormal modes. The Wronskian W[u H + , u ∞ ] has no real zeros, W[u H + , u ∞ ] = 0, however, it might very well have zeros in the lower half-plane with Im(ω) < 0. These zeros correspond to so-called quasinormal modes i.e. solutions of the form (1.9) which decay in time at an exponential rate. Note that quasinormal modes do not have finite energy on {t = const.}-slices (in particular they have infinite energy on Σ 0 = {t = 0}). However, they have finite energy for {t * = const.}-slices, where t * is a suitable time coordinate which extends regularly to the event horizon H R , see already (2.25). For a more precise definition, construction and a more detailed discussion of quasinormal modes in general we refer to [40]. Turning back to Kerr-AdS, we note that the bound (1.12) implies the existence of zeros of W[u H + , u ∞ ] exponentially close to the real axis as shown in [41], see also [97]. More precisely, it was shown that there exist axisymmetric quasinormal modes with frequencies m = 0 and (ω, ) = (ω n , n ) n∈N satisfying (1.14) While the previous results were proved in axisymmetry to simplify the analysis, in principle, they also extend to non-axisymmetric solutions as remarked in [41].
Semi-classical heuristics for distribution of quasimodes and quasinormal modes. We first turn to the heuristic distribution of the quasimode frequencies in the semi-classical (high frequency) limit. For large |m|, m ∈ Z, ≥ |m|, we expect a quasimode with frequencies m, , ω to exist, if the potentialṼ (r * , ω, m, λ m (aω)) appearing in the radial o.d.e. (1.10) satisfies (see Fig. 3) (r * , ω, m, λ m (aω)) π 2 l Figure 3: PotentialṼ with frequency ω, m, for which we expect quasimodes. The gray area is a suitable projection of the phase space volume.

2
(1. 15) should be an integer multiple modulo the Maslov index up to an exponentially small error. Thus, at least heuristically, we expect that for given but large |m|, ≥ |m|, there exist N (m, ) ∼ intervals of quasimodes with midpoint ω ∼ and length e − . While quasimode frequencies are defined through an open condition (c.f. (1.12)), quasinormal mode frequencies will be discrete and in a exponentially small neighborhood of quasimodes. Thus, we expect the quasinormal mode frequencies to be distributed as (1. 16) Refer to Fig. 4 for a visualization of the expected distribution of quasimodes and quasinormal modes.
For our heuristic analysis we will now consider a solution ψ of (1.2) which consists of an infinite sum of weighted quasinormal modes (Warning: A general solution cannot be written as a sum of quasinormal modes. where we require that the weightsã(m, , n) have superpolynomial decay. This ensures that the initial data (posed on a {t * = const.}-slice) are smooth where we assume that each individual quasinormal mode is suitably normalized. 2 Restricting this solution ψ to the event horizon yields for new coefficients a(m, , n) which satisfy |a(m, , n)| ∼ |ã(m, , n)u(r + , ω m n , m, )|. Now, note that the radial part of the quasinormal mode |u(r, ω m n , m, )| will be localized in the region of stable trapping, i.e. in the region {r * ≥ r * 2 } of Fig. 3. From semi-classical heuristics, we expect that only an exponentially damped proportion "tunnels" from the region of stable trapping through the barrier to event horizon at r = r + . More precisely, the damping factor of the exponent of |u(r + , ω m n , m, )| is expected to be proportional to Thus, choosing coefficientsã(m, , n) now corresponds to choosing coefficients a(m, , n) satisfying (1.20) and vice versa. In view of this, instead of choosingã(m, , n), we will go forward in our heuristic discussion by choosing coefficients a(m, , n) satisfying (1.20). The goal is to choose such coefficients such that ψ blows up at the Cauchy horizon!

Interior: Scattering from event to Cauchy horizon
We now turn to the interior problem. We will view some aspects of the propagation of ψ from the event horizon to the Cauchy horizon as a scattering problem as visualized in Fig. 5. We refer to [62] for a detailed discussion of the scattering problem on black hole interiors. Unlike in [62], we will not develop a full scattering theory for Kerr-AdS, but rather make use of a key insight from [62] adapted to our context. Recall from [62, Proposition 6.2] that on Reissner-Nordström-AdS, the scattering operator S H R →CH R in the interior has a pole at the frequency ω = 0, which is the characteristic frequency associated to the Killing generator of the Cauchy Figure 5: Interior scattering S H R →CH R from event horizon H R to Cauchy horizon CH R horizon T . In the present case for Kerr-AdS, it is the vector field K − := T + ω − Φ which generates the Cauchy horizon and thus the characteristic frequencies are those satisfying ω − ω − m = 0. For fixed frequency scattering, this means that the reflection coefficient R (i.e. the fixed frequency scattering operator from H R to CH R ) has a pole at ω − ω − m = 0 such that R is of the form There is a natural solution ψ defined in the black hole interior by continuing each quasinormal mode appearing in (1.17) into the interior. This solution is again smooth across H R and thus can be view as a solution arising from smooth data on a spacelike hypersurface which coincides with {t * = 0} on the exterior. Let us assume for a moment that the fixed frequency scattering theory also carries over to complex frequencies and that we can analytically continue the reflection coefficient R to the complex plane. We then expect that the continued solution ψ at the Cauchy horizon can be obtained by multiplying each individual coefficient ψ H as in (1.18) with the reflection coefficient R(ω m n , m, ). Moreover, neglecting r(ω m n , m, ) which is expected to be suitably bounded from below and above, and taking the L 2 (S 2 )-norm of the {u = const}-spheres on the Cauchy horizon CH R , formally yields where we recall that a(m, , n) decay exponentially as in (1.20). In order to resolve Conjecture 3, we have to determine whether for all coefficients a(m, , n) satisfying (1.20), the sum (1.22) remains uniformly bounded, or whether, for some choice of a(m, , n) satisfying (1.20), this sum is infinite. Before we address this issue in the next paragraph, we refer to Fig. 6 for an illustration of the main difference between the behavior of linear scalar perturbations on Reissner-Nordström-AdS and Kerr-AdS.

Small divisors and relation to Diophantine approximation
The convergence of (1.  The set R is Baire-generic and Lebesgue-exceptional. The set R can be written as a lim sup set as It is a countable intersection of open and dense sets such that R is of second category in view of Baire's theorem [2]. Thus, the set R is generic from a topological point of view, which we refer to as Baire-generic. On the other hand, from a measure-theoretical point of view, the set R is exceptional. Indeed, an application of the Borel-Cantelli lemma shows that the Lebesgue measure of R vanishes. This is the easy part of the famous theorem by Khintchine [63] stating that for a decreasing function φ, the set has full Lebesgue measure if and only if the sum q φ(q) diverges. Thus, R is Lebesgueexceptional .
More refined measure: The Hausdorff and packing measures. This naturally leads us to consider the more refined versions of measure, the so-called Hausdorff and packing measures H f , P f together with their associated dimensions dim H , dim P (see Section 2.1). The Hausdorff and packing measure generalize the Lebesgue measure to non-integers. In a certain sense, they can be considered to be dual to each-other: The Hausdorff measure approximates and measures sets by a most economical covering, whereas the packing measure packs as many disjoint balls with centers inside the set. While for all sufficiently nice sets these notions agree, they indeed turn out to give different results in our context.
We first consider the Hausdorff dimension. A version of the Borell-Cantelli lemma (more precisely the Hausdorff-Cantelli lemma) and using the natural cover for R shows that the set R is of Hausdorff dimension zero. This again can be seen as a consequence of a theorem going back to Jarník [58] and Besicovitch [4] which states the set W [φ] as in (1.26) has Hausdorff measure for s ∈ (0, 1). However, measuring also logarithmic scales, i.e. considering the Hausdorff measure H f for f = log t (r) for some t > 0, it follows that the set R is of logarithmic generalized Hausdorff dimension. On the other hand, using the dual notion of packing dimension, it turns out that R has full packing dimension, a consequence of the fact that it is a set of second category (Baire-generic) [29].
Summary of properties of R. To summarize, we obtain that • R is Lebesgue-exceptional, • R has zero Hausdorff dimension dim H (R) = 0, • R is of logarithmic generalized Hausdorff dimension, • R has full packing dimension dim P (R) = 1.
The above heuristics will enter in our revised conjectures, Conjecture 5 and Conjecture 6, which transcend Conjecture 3 and Conjecture 4 for Λ < 0. Before we turn to that in Section 1.5, we briefly discuss other aspects of PDEs and dynamical systems for which Diophantine approximation plays a crucial role.
Small divisors problems and Diophantine approximation in dynamical systems and PDEs. Most prominently, Diophantine approximation and the small divisors problem are intimately tied to the problem of the stability of the solar system [79] and more generally, the stability of Hamiltonian systems in classical mechanics. This stability problem was partially resolved with the celebrated KAM theorem [64,1,78] which roughly states that Lebesgue-generic perturbations of integrable Hamiltonian systems lead to quasiperiodic orbits. The small divisors problem and Diophantine approximation are ubiquitous in modern mathematics and arise naturally in many other aspects of PDEs and dynamical systems. We refer to [30,80] for a connection to wave equations with periodic boundary conditions and to the more general results in [43] as well as the monograph [90]. Similar results have been obtained for the Schrödinger equation on the torus in [65,59,28]. Further applications of Diophantine approximation include the characterization of homeomorphisms on S 1 by the Diophantine properties of their rotation numbers or analyzing the Lyapunov stability of vector fields, see the discussion in [67].

Conjecture 5 and Conjecture 6 replace Conjecture 3 and Conjecture 4 for Kerr-AdS
With the above heuristics in hand, we now transcend Conjecture 3 and Conjecture 4 for subextremal Kerr-AdS black holes with parameters below the Hawking-Reall bound in terms of the following two conjectures. We denote the set of all such parameters with P, see already (2.9).

Conjecture 5.
There exists a set P Blow-up ⊂ P of dimensionless Kerr-AdS parameters mass m = M √ −Λ and angular momentum a = a √ −Λ with the following properties • P Blow-up is Baire-generic (of second category), • P Blow-up is Lebesgue-exceptional (zero Lebesgue measure), and such that for every Kerr-AdS black hole with mass M = m/ √ −Λ and specific angular momentum a = a/ √ −Λ, where (m, a) ∈ P Blow-up , there exists a solution ψ to (1.2), which arises from smooth and compactly supported initial data (ψ 0 , ψ 1 ) on a suitable spacelike hypersurface with Dirichlet boundary conditions at infinity, and which blows up at the Cauchy horizon for every u ∈ R.
Remark 1.1. If there exist initial data (ψ 0 , ψ 1 ) leading to a solution ψ which blows up as in (1.28), this then shows that initial data (ψ 0 ,ψ 1 ) for which the arising solution does not blow up are exceptional in the sense that they obey the following co-dimension 1 property: The solution arising from the perturbed data (ψ 0 +cψ 0 ,ψ 1 +cψ 1 ) blows up for each c ∈ R\{0}. This is analogous to the notion of genericity used by Christodoulou in his proof of weak cosmic censorship for the spherically symmetric Einstein-scalar-field system [9,8]. Thus, Conjecture 5 gives a formulation of Conjecture 3. We note already that we will actually formulate in Remark 1.5 another more refined genericity condition for the set of initial data leading to solutions which blow up as in (1.28).
Remark 1.2. Note that in Conjecture 5 we have replaced the statement of blow-up in amplitude from Conjecture 3 with a statement about the blow-up of the L 2 (S 2 )-norm on the sphere. Indeed, the blow-up of the L 2 (S 2 )-norm in Conjecture 5, if true, implies that ψ L ∞ (S 2 ) (u, r) → +∞ as r → r − . In this sense, if Conjecture 5 is true, the amplitude also blows up. It is however an interesting and open question whether one may actually replace the L ∞ (S 2 ) blow-up statement in (1.28) with the pointwise blow-up for every (θ, φ * − ) ∈ S 2 . One may even speculate about the geometry of the set of (θ, φ * − ) ∈ S 2 for which pointwise blow-up holds. It appears that ultimately one has to quantitatively understand the nodal domains associated to the generalized spheroidal harmonics S m (aω − m, cos θ) at the interior scattering poles. • generalized Hausdorff dimension dim gH (P Blow-up ) = 1 + log, • full packing dimension dim P (P Blow-up ) = 2.
Moreover, in view of our discussion we additionally conjecture • P Bounded is Lebesgue-generic (full Lebesgue measure), and such that for every Kerr-AdS black hole with mass M = m/ √ −Λ and specific angular momentum a = a/ √ −Λ, where (m, a) ∈ P Bounded , all solutions ψ to (1.2), which arise from smooth and compactly supported initial data (ψ 0 , ψ 1 ) on a spacelike hypersurfaces with Dirichlet boundary conditions at infinity, remain uniformly bounded at the Cauchy horizon. Here, D(ψ 0 , ψ 1 ) is a (higher-order) energy of the initial data and C(m, a) is a constant depending on m and a.
(B) For all Kerr-AdS black holes with parameters in P, there exists a solution ψ to (1.2), which arises from smooth and compactly supported initial data on a spacelike hypersurfaces with Dirichlet boundary conditions at infinity and which blows up in energy at the Cauchy horizon. More general boundary conditions and Klein-Gordon masses. The above conjectures are both stated for Dirichlet conditions at infinity. Neumann conditions are also natural to consider and indeed well-posedness was proved in [96,57]. For Neumann conditions we also expect the same behavior as for the case of Dirichlet boundary conditions. For other more general conditions, it may be the case that linear waves grow exponentially (as for suitable Robin boundary conditions [57]) or on the other hand even decay superpolynomially as is the case for purely outgoing conditions [53]. For even more general boundary conditions, even well-posedness may be open.
In this paper we have focused on scalar perturbations satisfying (1.2). In particular, the choice of the Klein-Gordon mass parameter µ = 2 3 Λ ("conformal coupling") is the most natural as it arises from the linear scalar analog of (1.1) and also remains regular at infinity. However, in certain situations it may also be interesting to consider more general Klein-Gordon masses µ satisfying the Breitenlohner-Friedman [6] bound µ > 3 4 Λ. We also expect Conjecture 5 and Conjecture 6 to hold for Klein-Gordon masses for which the exterior is linearly stable, i.e. for µ > 3 4 Λ in the case of Dirichlet boundary conditions, and for 3 4 Λ < µ < 5 12 Λ together with additional assumptions in the case of Neumann boundary conditions [57].
Regularity of the initial data. We stated Conjecture 5 and Conjecture 6 for smooth (C ∞ ) initial data. One can also consider classes of initial data which are more regular (e.g. Gevrey or analytic) or less regular (e.g. Sobolev). From our heuristics, we expect that the analogs of Conjecture 5 and Conjecture 6 remain valid both for rougher data in some suitably weighted Sobolev space (see [54]) and more regular data of Gevrey regularity with index σ > 1 and analytic data (σ = 1). Only in the exceptional and most regular case of initial data with Gevrey regularity σ < 1 (note that this is more regular than than analytic) in the angular direction ∂ φ , we expect the analog of Conjecture 5 to break down. In particular, for axisymmetric data (or data supported only on finitely many azimuthal modes m), we expect the arising solution to remain uniformly bounded at the Cauchy horizon for all parameters in P.

Theorem 1: Conjecture 5 is true
Our main result is the following resolution of Conjecture 5.

Theorem 1. Conjecture 5 is true.
The proof of Theorem 1 will be given in Section 9.
Remark 1.5. In the proof of Theorem 1 we will not only construct a single solution which blows up leading to genericity of initial data as in Remark 1.1 but we will actually obtain what is perhaps a more satisfying genericity condition on the initial data which are smooth and of compact support. We formulate this condition in Corollary 1 in Section 9. Remark 1.6. We also prove in Section 9 the statement about the packing dimension of P Blow-up as conjectured in Remark 1.3. The statements concerning the Hausdorff dimension, however, remain open. Remark 1.7. In principle, our proof is expected to also apply to Neumann boundary conditions as well as to more general Klein-Gordon masses satisfying the Breitenlohner-Friedman bound [6] as discussed at the end of Section 1.5.

Outlook on Conjecture 6
We also expect that our methods provide a possible framework for the resolution of Conjecture 6.
First, note that the blow-up statement of Theorem 1 is strictly stronger than the H 1 loc blow-up conjectured in Conjecture 6(B). Thus, Theorem 1 shows that Conjecture 6(B) is true for black hole parameters in the set P Blow-up . For parameters not contained in P Blow-up , we expect that a quasinormal mode which decays at a sufficiently slow exponential decay rate compared to the surface gravity of the Cauchy horizon will blow up in energy at the Cauchy horizon. This would show Conjecture 6(B). Towards Conjecture 6(A), we note that our proof, particularly formula (9.6) of Proposition 9.1, reveals the main obstruction for boundedness. Together with the methods used in [61] for the Reissner-Nordström-AdS case, this can serve as a starting point for a resolution of Conjecture 6(A).

Turning the heuristics of Section 1.4 into a proof of Theorem 1
We will now outline how we turn our heuristics of Section 1.4 into a proof of Conjecture 5, i.e. Theorem 1.
We are interested in constructing a solution of (1.2), arising from smooth and compactly supported initial data, which blows up as in (1.28), if the dimensionless parameters m, a satisfy certain a non-Diophantine condition.
We remark that unlike in our heuristic discussion, we will not make use of quasinormal modes and the frequency analysis will be purely based on the real axis with ω ∈ R. Indeed, our approach can be interpreted as replacing quasinormal modes with quasimodes. This will also manifest itself in the fact that the roles of 1 ω−ω−m will be changed: In the heuristic analysis, we considered the quasinormal mode frequencies ω m n which are (complex) roots of the Wronskian W[u H + , u ∞ ] and the small divisors came from 1 ω m n −ω−m . In the actual proof of Theorem 1, we will instead consider the real frequencies ω = ω − m (i.e. the roots of ω − ω − m = 0) and as we will see, the small divisors will then appear from the Wronskian evaluated at the characteristic frequency In view of the distribution of the quasimode frequencies discussed in Section 1.2, this will lead to a (generalized) non-Diophantine condition which we will address in more detail further below.
Initial data and exterior analysis (Section 6 and Section 7). We begin our discussion with our choice of initial data. In Section 6 we will carefully impose smooth and compactly supported initial data Ψ 0 , Ψ 1 ∈ C ∞ c (Σ 0 ) for (1.2) on the spacelike hypersurface Σ 0 = {t = 0} which-with foresight-will be chosen to satisfy and We also recall that u ∞ (r, ω, m, ) is the solution to the radial o.d.e. adapted to the Dirichlet boundary condition at I. Complementing the data with vanishing data on H L ∪ B H , the data Ψ 0 , Ψ 1 define a solution on the black hole interior. Different from our heuristic discussion with quasinormal modes (i.e. fixed frequency solutions) in Section 1.4, in the present case, we do need to consider the analog of a "full" scattering operator S H R →CH R from the event horizon to the Cauchy horizon which would be of the form where F H and F CH represent (generalized) Fourier transforms along the event and Cauchy horizon, respectively. Thus, from the exterior, we need to determine the generalized Fourier transform F H [ψ H ] along the event horizon. Such a characterization in terms of the chosen initial data from above is the content of Section 7. While in the actual proof (see already Proposition 7.1), we will use a suitably truncated generalized Fourier transform, we may formally think of F H [ψ H ] as having the form We already remark that a consequence of the smoothness of the initial data is that G(Ψ 0 , Ψ 1 , m, ) decays superpolynomially in m and , cf. (6.26) in Section 6.
Interior analysis (Section 8, Section 9 and some of Section 3). Turning to the interior analysis, we recall from our heuristic discussion in Section 1.3 that the analog of the scattering operator (1.36) from the event to the Cauchy horizon has poles at the characteristic frequencies ω − ω − m = 0 with respect to K − . In our heuristic discussion in Section 1.4 based on quasinormal modes and fixed frequency scattering, these poles formally lead to (1.22). In the actual proof, based on frequency analysis on the real axis, the scattering poles become evident in formula (9.6) stated in Proposition 9.1 which roughly translates to the statement that, as r → r − , we have where Err(D) is uniformly bounded by an (higher order) energy of the initial data. Combining the exterior with the interior: Occurrence of small divisors and the proof of Theorem 1 (Section 9). We will now connect the exterior analysis to the interior. In particular, formally plugging (1.37) into (1.38) and noting that where we also used that the (renormalized) reflection coefficient satisfies |r(ω = ω − m, m, )| ∼ |m| which we will show in Lemma 8.6. Also recall from before that the error term |Err(D)| is shown to remain uniformly bounded as r → r − . Remark that in the actual proof we will not quite show (1.41) but rather obtain (9.44) which corresponds to (1.41) in a certain limiting sense. We also recall from the discussion of the exterior analysis that the term |m| 2 |G(Ψ 0 , Ψ 1 , m, )| 2 which appears in the sum of (1.41) as the numerator, decays superpolynomially in m and . Thus, at least formally, in order to show blow-up for (1.41), it is necessary that small divisors in (1.41) occur infinitely often, i.e. that the Wronskian evaluated at the interior scattering poles 2) decays (at least) superpolynomially for infinitely many (m, ). In our proof, we will actually require from the black hole parameters m, a that this Wronskian decays exponentially Before we address the validity of (1.42), we will assume for a moment that indeed the black hole parameters m, a are such that (1.42) holds true. Then, explicitly choose that the subsequences m i and i in (1.33) coincide with the infinite sequences which fulfill (1.42). Then, we formally obtain the blow-up result of Theorem 1 as Similarly to the remark before, in reality, (1.43) holds true only in a certain limiting sense, cf. (9.44)-(9.46) of Section 9. Already from (1.43) and (1.42) we obtain the following genericity condition on the initial data leading to blow-up. This will be formulated as Corollary 1 in Section 9.
The non-Diophantine condition and its relation to quasimodes (Section 4, Section 5, and some of Section 3). Finally, this leaves us to address the question of whether the small divisors in (1.41) actually appear infinitely often, more precisely, whether (1.42) holds true. The condition (1.42) constitutes a generalized non-Diophantine condition on the black hole parameters m, a in view of its relation to the (discrete) Bohr-Sommerfeld quantization conditions from our heuristic discussion Section 1.2. In our actual proof, the non-Diophantine condition which we define in Definition 5.3 in Section 5 is more technical than (1.42), though (1.42) should be considered as its key property. We denote the set of dimensionless black hole parameters m, a which satisfy the condition with P Blow-up . The statement that is P Blow-up is Baire-generic but Lebesgue-exceptional is the content of Section 5.2 and Section 5.3, respectively. Both proofs crucially rely on estimates developed in Section 4.
Connecting to the discussion of quasimodes before, we note that the non-Diophantine condition of (1.42) can be interpreted as the statement that there exist infinitely many quasimodes with frequency ω = ω − m. This also implies that there exist infinitely many quasinormal modes with (complex) frequencies ω exponentially close to ω = ω − m. However, note that quasimodes are more robust to perturbations in the sense that if ω, m, are frequencies of a quasimode, there exists a (exponentially small) neighborhood of ω such that for eachω in that neighborhood, the frequenciesω, m, would also describe a quasimode. It is also this robustness which is a key advantage of quasimodes over an approach based on quasinormal modes as in the heuristic discussion in Section 1.4.

Outline of the paper
In Section 2 we set up the Kerr-AdS spacetime, recall the well-posedness of (1.2) and the decay statement for solutions on the exterior. We also introduce Carter's separation of variables. Section 3 is devoted to the analysis of the angular o.d.e. In Section 4 we analyze the radial o.d.e. on the exterior and introduce suitable solutions of the radial o.d.e. associated to trapping at the interior scattering poles ω = ω − m. With the estimates from Section 4 in hand, we define the set P Blow-up in Section 5 and show its topological and metric properties.
Then, for arbitrary but fixed parameters p ∈ P Blow-up we define suitable compactly supported and smooth initial data in Section 6. In Section 7 we treat the exterior problem and conclude with a representation formula of the solution along the horizon in terms of the initial data. In Section 8 we first show suitable estimates for solutions of the radial o.d.e. in the interior before we finally conclude the paper with the proof of Theorem 1 in Section 9.

Acknowledgements
The author would like to express his gratitude to his advisor Mihalis Dafermos for his support and many valuable comments on the manuscript. The author also thanks Harvey Reall, Igor Rodnianski and Yakov Shlapentokh-Rothman for insightful discussions and helpful remarks. This work was supported by the EPSRC grant EP/L016516/1. The author thanks Princeton University for hosting him as a VSRC.

Hausdorff and Packing measures
We begin by introducing the Hausdorff and packing measure. We refer to the monograph [29] for a more detailed discussion. For an increasing dimension function f : If f (r) = r s , we write H s = H r s and for s ∈ N, the measure H s reduces to the Lebesgue measure up to some normalization. While the Hausdorff measure quantifies the size of a set by approximation it from outside via efficient coverings, we also recall the dual notation: The packing measure quantifies the size of sets by placing as many disjoint balls with centers contained in the set. Again, for a dimension function f , we define the pre-measure and finally the packing measure as

Hausdorff and Packing dimensions
For f (r) = r s Hausdorff and Packing dimensions dim H and dim P are now characterized as the jump value, where the respective measure jumps from 0 to ∞, more precisely We also say that a set A has generalized Hausdorff dimension dim gH (A) = s + log if the jump appears for the dimension function f (r) = r s log t (r) for some t > 0.

Parameter space
We let the value of the cosmological constant Λ < 0 be fixed throughout the paper. For convenience and as it is convention, we re-parametrize the cosmological constant by the AdS radius We consider Kerr-AdS black holes which are parameterized by their mass M > 0 and their specific angular momentum a = 0. Moreover, without loss of generality we will only consider a > 0 and require 0 < a < l for the spacetime to be regular. For M > 0, 0 < a < l, we consider the polynomial We are interested in spacetimes without naked singularities. To ensure this, we define a parameter tuple (M, a) ∈ R 2 >0 to be non-degenerate if 0 < a < l and ∆(r) defined in (2.7) has two real roots satisfying 0 < r − < r + . Finally, to exclude growing mode solutions (see [27]) we assume the Hawking-Reall (non-superradiant) bound r 2 + > al. (2.8) This leads us to the definition of the dimensionless black hole parameter space Note that in view of (2.6), On the parameter space P, we will also use the global coordinates (ϑ, a), where (2.10) Thus, for each a, there exists an interval (ϑ 1 (a), ϑ 2 (a)) and a smooth embedding (ϑ 1 (a), ϑ 2 (a)) → P, ϑ → (m(ϑ), a) which also depends smoothly on a. We define the vector field Γ on P by in coordinates (ϑ, a). We define Φ Γ τ as the flow generated by Γ. Finally, remark that P is a Baire space as a (non-empty) open subset of R 2 . In particular, this allows us to speak about the notion of Baire-exceptional and Baire-generic subsets. Recall that a subset is Baire-exceptional if it is a countable union of nowhere dense sets and a subset is called Baire-generic if it is a countable intersection of open and dense sets. Note that if a subset is Baire-generic then its complement is Baire-exceptional and vice versa. Finally, in a Baire space every Baire-generic subset is dense.

Kerr-AdS spacetime
Fixed manifold. We begin by constructing the Kerr-AdS spacetime. We define the exterior region R and the black hole interior B as smooth four dimensional manifolds diffeomorphic to R 2 × S 2 . On R and on B we assume to have global (up to the well-known degeneracy on S 2 ) coordinate charts These coordinates (t, r, φ, θ) are called Boyer-Lindquist coordinates. If it is clear from the context which coordinates are being used, we will omit their subscripts throughout the chapter.
The Kerr-AdS metric. For (m, a) ∈ P and M = ml/ √ 3 and a = al/ √ 3, we define the Kerr-AdS metric on R and B in terms of the Boyer-Lindquist coordinates as where and ∆ is as in (2.7). We will also write ∆ x := 1− a 2 l 2 x 2 which arises from the substitution x = cos θ in ∆ θ . We also define (2.16) Now, we time-orient the patches R and B with −∇t R and −∇r B , respectively. We also note that ∂ t and ∂ φ are Killing fields in each of the patches. The inverse metric reads On R and B, we define the tortoise coordinate r * (r) by where ∆ is as in (2.7). For definiteness we set r * (r = +∞) := π 2 l on R and r * ( 1 2 (r + + r − )) = 0 on B.
Eddington-Finkelstein-like coordinates. We also define Eddington-Finkelstein-like coor- In these coordinates the spacetime (R, g KAdS ) can be extended (see [54] for more details) to a time-oriented Lorentzian manifold (D, g KAdS ) defined as D : We identify these regions and denote the (right) event horizon as H R := {r = r + }. The Killing null generator of the event horizon is (2.20) The Killing field K + is called the Hawking vector field and is future-directed and timelike in R, a consequence the Hawking-Reall bound r + > al.
To attach the (left) Cauchy horizon CH L we introduce in B further coordinates (v, r, θ,φ − ), as In these coordinates, the Lorentzian manifold extends smoothly to r = r − and the null hypersurface CH L := {r = r − } is the left Cauchy horizon with null generator on B and attach the right Cauchy horizon CH R as CH R = {r = r − } in this coordinate system. Indeed, K + and K − extend to Killing vector fields expressed as They are past directed Killing generators of H L and CH R , respectively. Finally, we attach the past and future bifurcation spheres B H and B CH . Formally, they are defined as Finally, we define the Cauchy horizon CH := CH L ∪ CH R ∪ B CH . This is standard and we refer to the preliminary section of [20] for more details. The metric g KAdS extends to a smooth Lorentzian metric on B H , B CH and we define (M KAdS , g KAdS ) as the Lorentzian manifold constructed above. Moreover, T := ∂ t and Φ := ∂ φ extend to smooth Killing vector fields on M KAdS with K Kerr-AdS-star coordinates. On the exterior region R we define an additional system of coordinates (t * , r, θ, φ * ) from the Boyer-Lindquist coordinates through and dB dr = aΞ ∆ and A = B = 0 at r = +∞. As shown in [54, Section 2.6], these coordinates extend smoothly to the event horizon H R and we call the coordinates (t * , r, θ, φ * ) covering R ∪ H R Kerr-AdS-star coordinates. Note that the event horizon is characterized as H R = {r = r + } in these coordinates.
Foliations and Initial Hypersurface. We foliate the region R ∪ H R with constant t * hypersurfaces Σ t * which are spacelike and intersect the event horizon at r = r + . For the initial data we will consider the axisymmetric spacelike hypersurface (2.26) Note that Σ 0 does not contain the bifurcation sphere B H . We will impose initial data on Σ 0 ∪B H ∪ H L . We will choose the support of our initial data to lie in a compact subset K ⊂ Σ 0 ∩ {r ≥ 2r + }. Thus, we assume vanishing data on H L ∪ B H . This will be made precise in Section 6.
Boundary conditions. Note that the conformal boundary I, expressed formally as {r = +∞}, is timelike, as a consequence, (M KAdS , g KAdS ) is not globally hyperbolic. Thus, in addition to Cauchy data for (1.2), we will also impose Dirichlet boundary conditions at I = {r = +∞}.

Conventions
If X and Y are two (typically non-negative) quantities, we use X Y of Y X to denote that X ≤ C(M, a, l)Y for some constant C(M, a, l) > 0 depending continuously on the black hole parameters (M, a, l), unless explicitly stated otherwise. We also use X = O(Y ) for |X| Y . We use X ∼ Y for X Y X and if the constants appearing in , , ∼ or O depend on additional parameters a i we include those as subscripts, e.g. X a1a2 Y .
In Section 6 we will fix parameters (m, a) ∈ P Blow-up and all constants appearing in and throughout Section 6 Section 7, Section 8 will only depend on this particular choice and on l > 0 as in (2.6).
Further, we denote the total variation of a function f :

Norms and energies
To state the well-posedness result of (1.2) and the logarithmic decay result on the Kerr-AdS exterior, we define the following norms and energies in the exterior region R ∪ H R . These are based on the works [52,54,56], where more details can be found. In the region R ∪ H R we let / g and / ∇ be the induced metric and induced connection of the spheres S 2 t * ,r of constant t * and r. For a smooth function ψ we denote | / ∇ . . . / Now, we define energy densities in Kerr-AdS-star coordinates as and analogously for higher order energy densities. Here, (Ω i ) i=1,2,3 denote the angular momentum operators on the unit sphere in θ, φ * coordinates. We also define the energy norms on constant t * hypersurfaces as

Well-posedness and log-decay on the exterior region
In the following we state well-posedness for (1.2) and decay solutions with Dirichlet boundary conditions. The following theorem is a summary of results by Holzegel, Smulevici and Warnick shown in [52,54,56,57]. Theorem 2 ([52,54,56,57]). Let the initial data Ψ 0 , Ψ 1 ∈ C ∞ c (Σ 0 ). Assume Dirichlet boundary conditions at I and vanishing incoming data on H L ∪ B H . Then, there exists a unique solution AdS for every k ∈ N. We also have boundedness of the energy for t * 2 ≥ t * 1 ≥ 0 as well as for all higher order energies. Similarly, the energy along the event horizon is bounded by the initial energy as Moreover, the energy of ψ decays for all t * ≥ 0 and similar estimates hold for all higher order energies. Similarly, by commuting and applications of the Soboelv embeddings, ψ and all its derivatives also decay pointwise The wave equation (1.2) is formally separable [7] and we can consider pure mode solution in the Boyer-Lindquist coordinates of the form for two unknown functions u(r) and S m (aω, cos θ). Plugging this ansatz into (1.2) leads to a coupled system of o.d.e's. The angular o.d.e. is the eigenvalue equation of the operator P (aω) which reads The operator (2.38) is realized as a self-adjoint operator on a suitable domain in L 2 ((0, π); sin θdθ). where := d dr * . We also use the notationṼ := V − ω 2 , where the potential V is given by with purely radial part and frequency dependent part (2.42) We will be particularly interested in the case for which the frequency ω coincides with the interior scattering poles, i.e. ω = ω − m. Moreover, in order to be in the regime of stable trapping on the exterior we also want |ω| and |m| to be large. Hence, we will think of 1 m as a small semiclassical parameter. In particular, setting ω = ω − m in (2.39) and separating out the m 2 we obtain where V 1 is as in (2.41) and We begin our analysis with the angular o.d.e. (2.37) in the following Section 3.
For the operator P (ξ) as in (2.38) we change variables to x = cos θ. This is a unitary transformation and thus, the eigenvalues of P (ξ) are equal to the eigenvalues of P x given by The Sturm-Liouville operator P x is realized as a self-adjoint operator acting on a domain D ⊂ L 2 (−1, 1) which can be explicitly characterized as Having the same spectrum as P , the operator P x has eigenvalues (λ m (ξ)) ≥|m| with corresponding real-analytic eigenfunctions S m = S m (ξ, x) which satisfy We note that for ξ = a = 0, the eigenvalues (λ m ) ≥|m| reduce to the eigenvalues of the Laplacian on the sphere λ m (a = ξ = 0) = ( + 1). We also define the shifted eigenvalues in the sense of self-adjoint operators acting on D ⊂ L 2 (−1, 1). Hence, Having recalled basic properties of the angular problem we now focus on the interior scattering poles ω = ω − m for large m. In particular, we will only consider m = 0 for the rest of Section 3.

Angular potential W 1 at interior scattering poles in semi-classical limit
In the current Section 3.1 and in the following Section 3.2 we will consider the operator with corresponding eigenvalues λ m := λ m (aω − m). We re-write the eigenvalue problem asP 10) and In the semi-classical limit m 2 → ∞ we consider P error as a perturbation and W 1 determines the leading order terms of the eigenvalues and eigenfunctions. Consequently, our analysis focuses on W 1 which we analyze in the following lemma.
Lemma 3.1. Let W 1 be the angular potential defined in (3.12).
Proof. We start by expanding W 1 and obtain and We also note that We now consider the caseλ ≥ Ξ 2 and remark that We look at two cases now, a 2 ≥ 0 and a 2 < 0. If a 2 ≥ 0, then we directly infer that dW1 dx ≥ 2a 1 x. If a 2 < 0, then we use that x 3 < x and estimate Now, a direct computation yields Note that this shows (3.14) for x ∈ [0, 1] and we conclude 3. Together with (3.18), this also shows that W 1 (x) > 0 for x ∈ (0, 1] andλ = Ξ 2 such that we have 2. Finally forλ < Ξ 2 , we have W 1 > 0 everywhere because for each fixed x ∈ [0, 1), the functioñ λ → W 1 (x) is strictly decreasing and W 1 (x = 1) = Ξ 2 > 0. for some irrelevant normalization and further assume that for some large c > 0,

Angular eigenvalues at interior scattering poles with
• the error-control function satisfies Then, there exists an error function ϑ satisfying |ϑ| f,g u −2 such that for all u sufficiently large the following holds true. There exists a bound state w (i.e. a solution which is recessive at both ends x → ±∞) of the differential equation

23)
if and only if there exists a positive integer n ∈ N such that 2 π x0 x0 −f dx + ϑ = 2n + 1 u . (3.24) With the above proposition in hand we proceed to the main proposition of this subsection, where we recall that we still consider the case ω = ω − m.
Proposition 3.2. Let p 0 ∈ P be arbitrary but fixed. Then, for almost everyλ 0 ∈ (Ξ 2 , ∞) (more precisely, for everyλ 0 ∈ (Ξ 2 , ∞) \ N p0 for some Lebesgue null set N p0 ), there exists a strictly increasing sequence of natural numbers (m i ) i∈N such that for every i ∈ N, the operator P ω− admits an eigenvalue λ i := λ mi i = λ mi i (ω = ω − m) satisfying where |λ We consider the formulation of the angular o.d.e. in (3.9) and moreover change coordinates This yields the equivalent eigenvalue problem − d 2 dy 2 g + (m 2 W 1 + P error )g = 0 (3.28) for g in a dense domain of L 2 (R, w(y)dy) with weight From Lemma 3.1 we have that W 1 has a unique positive root forλ > Ξ 2 which we denote with y 0 (λ) := y(x 0 (λ)). We also define where we recall that W 1 is symmetric around the origin. For the potential W 1 , we have (e.g. [31, p. 118]) that ξ : (Ξ 2 , +∞) → R,λ → ξ(λ) is a strictly increasing smooth (even real-analytic) function. Further note that so by the inverse function theorem, ξ has a smooth inverse. By a standard result on Diophantine approximation (see e.g. [44, Theorem 6.2]), we have that for each x ∈ R \ N , where N is a Lebesgue null set, there exist sequences of natural numbers (n i ) i∈N and (m i ) i∈N with n i+1 > n i and m i+1 > m i such that for all i ∈ N. Now, since ξ has a smooth inverse, there exists a Lebesgue null set N p0 for a sequence of natural numbers (n i ) i∈N and (m i ) i∈N with n i+1 > n i and m i+1 > m i . Now, we will apply Proposition 3.1. First, we see that for allλ in a small neighborhood of λ 0 ∈ (Ξ 2 , ∞) \ N p0 , the potential W 1 (y) and P error satisfy the assumptions of Proposition 3.1: Indeed, both W 1 and P error are smooth. Moreover, W 1 has two simple roots which do not coalesce and y 0 |W 1 |dỹ diverges as y → ±∞. Finally, for |y| → ∞, we have as well as Thus, we infer for some c > 0 sufficiently large, where From Proposition 3.1 we now conclude that the eigenvalues λ =λm 2 forλ in a neighborhood of λ 0 are characterized by for n ∈ N, where |ϑλ 0 | λ 0 m −2 . Now, for fixedλ 0 ∈ (Ξ 2 , ∞) \ N p0 , let the sequence (m i , n i ) i∈N as above be such that (3.33) holds. Then, we obtain associated eigenvalues from (3.38) which satisfỹ The last equality holds due Taylor's theorem and (3.31).

Bounds on ∂ ξ λ m and ∂ ξ S m near interior scattering poles
In the proof of Theorem 1 in Section 9 we will need to control the quantities ∂ ω λ m (aω) and ∂ ω S m (aω) near the interior scattering poles, i.e. for ω ≈ ω − m. We will choose our initial data in Section 6 to be supported on angular modes m > 0 which are large and positive. Thus, for the rest of this subsection, we assume that m > 0 and think of 1/m as a semiclassical parameter. We first note that a direct computation shows that solves the inhomogeneous o.d.e.
with Dirichlet boundary conditions at x = ±1, where We will first consider ∂ ξ λ m .
and thus, Proof. Taking the L 2 -inner product of (3.41) with S m and using that P x is self-adjoint, shows that from which we obtain in view of S m , S m L 2 (−1,1) = 1. Now, the claim follows from the fact that ∂ ξ P x |ξ|+|m|.
It is more difficult to obtain estimates for ∂ ξ S m which we express as where is the inhomogeneous term of (3.41), Res(λ; P x ) is the resolvent and Π ⊥ S m is the orthogonal projection on the orthogonal complement of S m . At this point we also remark that both ∂ ξ S m and H are orthogonal to S m which follows from ξ → S m , S m L 2 (−1,1) = 1 and (3.45), respectively.
A possible way to control the resolvent operator Res(λ m ; P x )Π ⊥ S m is to show lower bounds on the spectral gaps |λ m, (aω) − λ m, +1 (aω)| uniformly in m, → ∞ and ω ≈ ω − m. Our approach is based on an explicit construction of the resolvent kernel via suitable approximations with parabolic cylinder functions and Airy functions.
We begin by noting that from standard results on solutions to Sturm-Liouville problems, each eigenfunction S m is either symmetric or anti-symmetric around x = 0. If S m is antisymmetric around x = 0 we have S m (x = 0) = 0, i.e. Dirichlet boundary conditions at x = 0. Analogously, if S m is symmetric, we have Neumann boundary conditions at x = 0, i.e. d dx S m (x = 0) = 0. Also note that ∂ ξ S m inherits the symmetry properties of S m . Hence, the problem reduces to studying the interval x ∈ [0, 1) with Dirichlet/Neumann boundary conditions at x = 0 and Dirichlet boundary conditions at x = 1. In view of the above, ∂ ξ S m will satisfy depending on S m (x = 0) = 0 or d dx S m (x = 0) = 0, respectively, as well as In addition to satisfying the above boundary conditions, ∂ ξ S m is also a solution of the inhomogenous o.d.e. (3.41) which we explicitly write out as where | | < 1 is such that ξ = amω − + m . Moreover, ∂ ξ S m and H admit the same symmetries as S m such that, both H and ∂ ξ S m are orthogonal to S m in L 2 ([0, 1)). Also recall that As in the proof of Proposition 3.2, we introduce the variable y = y(x) through the conditions as well as the associated Hilbert space This can be computed explicitly as Note that (3.57) In this new variable, we define Then, we re-write (3.51) as We recall the definition of W 1 in (3.12) as and define as well as (3.63) where we recall that s p satisfies Dirichlet/Neumann boundary conditions at y = 0 and vanishes at y = +∞. We also note that the previous orthogonality properties remain, i.e. both s p and w −1 F s 1 are orthogonal to s 1 in the Hilbert space L 2 ([0, ∞), w(y)dy).
In order to construct the resolvent operator, we will first state the existence of a further suitable solution s 2 to the homogeneous equation which is linearly independent from s 1 . This is the content of the following lemma which will be proved in Section 3.4. Proof. This is proved in Section 3.4, more specifically the claim follows from Lemma 3.13 and Lemma 3.18.
With s 2 in hand we will now construct the integral kernel of the resolvent Res(λ m ; P x )Π ⊥ S m in y-coordinates. More specifically, we show  for some constants c p1 , c p2 ∈ R. It remains to show that c p2 = 0 and we consider the cases of Dirichlet/Neumann conditions of s 1 at y = 0 independently.

Semi-classical resolvent estimates near interior scattering poles
Throughout this subsection (Section 3.4) we assume that and m > 0. The goal of this subsection is to show Lemma 3.3. We first argue that for sufficiently large m, we only need to consider the caseλ > Ξ 2 as all eigenvalues λ m (aω − m) at the interior scattering poles are larger than Ξ 2 m 2 . Proof. This is immediate as forλ ≤ Ξ 2 and sufficiently large m, the operator − d 2 dy 2 + m 2 W 1 + W 2 is strictly positive in view of Lemma 3.5. Thus, it suffices to show Lemma 3.3 forλ > Ξ 2 and we consider the caseλ ∈ (Ξ 2 , Ξ 2 + 1] in Section 3.4.1 and the caseλ ∈ (Ξ 2 + 1, ∞) in Section 3.4.2.

The case Ξ
Letλ ∈ (Ξ 2 , Ξ 2 + 1]. In this range,λ can be arbitrarily close to Ξ 2 . Asλ → Ξ 2 , the root y 0 > 0 of the potential W 1 (y) coalesces with y = 0. Thus, our estimates need to be uniform in this limit and the appropriate approximation will be given by parabolic cylinder functions. To do so we will introduce the following Liouville transform which is motivated by [82]. We define a new variable 4 ξ = ξ(y) (3.77) to satisfy dξ dy where we choose α > 0 such that ξ(y 0 ) = α > 0 and ξ(y = 0) = 0. By construction, this defines ξ = ξ(y) as a smooth (even real-analytic) increasing function with values in [0, ∞), see also [82,Section 2.2]. Note that this holds true as the right hand side satisfies for y > 0. Equivalently, the function ξ(y) can be expressed as We also consider y = y(ξ) as a function ξ and define where we recall that s 1 was defined in (3.58). In this new variable ξ, the function where the error function Ψ is given by (3.84) Since W 1 is analytic and non-increasing inλ, we apply [82, Lemma 1] to conclude that Ψ is continuous for (ξ,λ) ∈ [0, ∞) × [Ξ 2 , Ξ 2 + 1]. Now, we define the error-control function (see (6.3) of [82]) with Ω(x) = |x| 1 3 . We will now bound the total variation of the error-control function F 1 in (3.85). To do so we first show Lemma 3.7. The smooth and monotonic functions ξ = ξ(y) and y = y(ξ) as defined in (3.78) satisfy for all ξ sufficiently large.
where we have used that for ξ sufficiently large. Hence, for ξ sufficiently large. Recall that Ψ is continuous everywhere and Ω = |x| Having controlled the error terms we now proceed to the definition of our fundamental solutions based on appropriate parabolic cylinder functions. Proposition 3.4. There exist solutions w 1 and w 2 of (3.83) satisfying where U andŪ are parabolic cylinder functions defined in Definition A.2 in the appendix. The error terms satisfỹ where w 1 is as in Proposition 3.4 and A 1 = 0 is a real constant.
Proof. Both functions σ 1 and w 1 are non-trivial solutions to (3.83) which are recessive as ξ → ∞ (y → ∞). The claim follows now as the space of solutions of (3.83) which are recessive as ξ → ∞ is one-dimensional.
Using the parabolic cylinder functions, we now define a solution σ 2 which is linearly independent of σ 1 . Here, W y and W ξ denote the Wronskians with respect to the y and ξ variable.
Proof. We estimate using (A.22) that and  Proof. We plug (3.117) into the left hand side of (3.118) and we will estimate both terms independently.
For the first term, we change variables from y to ξ, use that x → E U (b, x) is non-decreasing, as well as Lemma 3.12 Now, we use the bounds on M U and W(w 1 , w 2 ) from Proposition A.1 and Lemma 3.9 to deduce where we used the Cauchy-Schwarz inequality and the fact that s 1 satisfies (3.59) as well as (3.44).
For the second term we argue similarly and obtain  Recall also that s 1 as defined in (3.58) is a solution of (3.124). As before, we define y 0 as the unique non-negative root ofW 1 (y). It satisfies which becomes arbitrarily large forλ → ∞. Our estimates will be uniform in this limit.
Term I. We estimate term I as We consider the first term appearing in (3.142) and in view of Lemma 3.14 we obtain For the second term involving the second derivative, we use (3.126) to conclude that For the third term we use that |W 2 | 1 − x(y) 2 such that For the last term in (3.142), we have Term II. For this term, we use Taylor's theorem around the point y = y 0 and a lengthy but direct computation shows that uniformly in y ∈ [y 0 − 1, y 0 + 1] from which we conclude that |II| λ 1 2 . Term III. In the region y ∈ [y 0 + 1, ∞) we first have such that which is integrable at y = ∞. Moreover, we have Combining the estimates (3.149) and (3.150) we obtain that |III| λ 1 2 which concludes the proof.
Finally, we also introduceŴ which we will bound from below in the following. Proof. First, for y 0 − 1 ≤ y ≤ y 0 we have and for y 0 ≤ y ≤ y 0 + 1 we have where we have used the monotonicity ofW 1 . Hence, for |y − y 0 | ≤ 1. Now, using M Ai (x) 1 we conclude that for |y − y 0 | ≤ 1 we have , Moreover, the Wronskian of w 1 and w 2 satisfies  Proof. Note that both, w 1 and s 1 are recessive as y → ∞. Since the space of solutions which are recessive at y → ∞ is one-dimensional, we conclude that s 1 and w 1 are linearly dependent.
In view of Lemma 3.17 we define where w 2 is as in (3.159). Note that this implies that W(s 1 , s 2 ) = 1. Proof. Analogous to the proof of Lemma 3.13 we first estimate Now, plugging these estimates into (3.166) and using that E −1 Ai (λ 1 3 ς(y)) is a decreasing function, we conclude (3.169) For the second term, we argue similarly and estimate (3.170) The radial o.d.e. on the exterior We will now derive for which frequency parameters (ω, m, ) the poles of the interior scattering operator at ω = ω − m coincide with frequency parameters which are exposed to stable trapping on the black hole exterior. This allows us then to define the set P Blow-up in Section 5.1. Thus, we will analyze the radial o.d.e. at frequency ω = ω − m.

Resonance: Radial o.d.e. at interior scattering poles allows for stable trapping
We will first determine the range of angular eigenvalues λ m (amω − ) at the interior scattering poles ω = ω − m for which the radial o.d.e. admits stable trapping. Recall from (2.44) that the normalized high frequency part of the potential with ω = ω − m is given by where r cut := 3M Ξ . For the potential V main to admit a region of stable trapping, we require that V main has two roots r 1 < r 2 , see already Fig. 7. A sufficient condition for that is that the angular eigenvalues λ m (aω − m)m −2 are such that (4.3) is negative and (4.4) is positive. In this case, we denote with r 1 = r 1 (λ m (aω − m)m −2 , p) < r 2 = r 2 (λ m (aω − m)m −2 , p) the two largest roots. (We will show later that indeed, these are the only roots for r ≥ r + .) This leads us to define the following range of angular eigenvalues Remark that the last condition in (4.6) will be used in Lemma 5.2 while the other conditions in (4.6) guarantee that V main has two roots. Here, we also recall the definition of Γ = ∂ ∂ϑ in (2.11), where ϑ = aω − .
By construction, E p is a bounded interval and we define µ 0 (p) := inf E p , µ 1 (p) := sup E p . We will show that E is a fiber bundle. To do so we first show that E p is open and non-empty. Proof. First, we will show that which in turn follows from To see that r 2 − < al holds true, we write ∆(r) in terms of r − as Hence, implies 3r 4 − < a 2 l 2 (4.11) from which r 2 − < al (4.12) follows. Moreover, a direct computation shows that for µ < ω 2 − (l 2 − a 2 ) + 2aω − Ξ and ω 2 − (l 2 − a 2 ) + 2aω − Ξ − µ sufficiently small, we have Finally, note that r 2 (µ, p) → +∞ as µ → ω 2 − (l 2 −a 2 )+2aω − Ξ from below. Hence, the claim follows from the fact that to highest order in r, the last condition in (4.6) is fulfilled. More precisely, (4.14) From (4.6) it also follows that µ 0 and µ 1 are manifestly continuous functions on P. Hence, E is a (topological) fiber bundle. Now, also note that E is trivial with global trivialization ϕ E : E → P × (0, 1), (p, µ) → (p, µ µ1−µ0 − µ0 µ1−µ0 ) and we find (using this trivialization) two global sections σ 1 ∈ Γ(E) and σ 2 ∈ Γ(E) with σ 1 (p) < σ 2 (p) (4.15) for all p ∈ P (in mild abuse of notation). For definiteness, we take Having constructed σ 1 and σ 2 , we will now show the existence of exactly two turning points r 1 < r 2 of V main .
Proof. By construction of σ 1 and σ 2 , for any m −2 λ m ∈ (σ 1 , σ 2 ), the potential V main has a maximum and satisfies lim r→∞ V main < 0, V main (r = r + ) < 0 and V main (r = r cut ) > 0, (4.19) where r cut = 3M Ξ . See also [56, Lemma 3.1]. We will show now that V main has exactly two roots in [r + , ∞) from which (4.17) and (4.18) follow. Indeed, in view of the above, V main either has two or four roots in [r + , ∞). To exclude the case of four roots, it suffices to exclude the case of three critical points in [r + , ∞). To see this, note that has at most three real roots, one of which is in [r + , ∞) in view of the construction above. Indeed, one other root has to lie in (−∞, r − ] as

Fundamental pairs of solutions associated to trapping
We will now define various solutions to the radial o.d.e. associated to the boundary and to the turning points. Note that the turning points define the transition from the trapping region to the semiclassically forbidden region.

Solutions associated to the boundary
We first define the associated solution to the radial o.d.e. (2.39) which satisfies the Dirichlet boundary conditions at r * = π 2 l. Definition 4.1. For all frequencies ω ∈ R, m ∈ Z, ∈ Z ≥|m| we define the solution u ∞ as the unique solution to (2.39) satisfying where we recall that = d dr * . Definition 4.2. For all frequencies ω ∈ R, m ∈ Z, ∈ Z ≥|m| we also define u H + and u H − as the unique solutions to (2.39) satisfying

Solutions associated to turning points at interior scattering poles
For the solutions associated to the turning points we only consider the radial o.d.e. for ω = ω − m.
We define solutions associated to the turning points r * 1 and r * 2 as illustrated in Fig. 7. In view of Definition 4.3. Assume that σ 1 ≤ λ m m −2 ≤ σ 2 and denote the turning points of V main with r * 1 := r * (r 1 ) < r * (r 2 ) =: r * 2 . Then, for some fixed (p) > 0, define (4.27) (4.28) and the error control functions Then, the error control functions H 1 and H 2 satisfy Proof. We begin with H 2 . From we have that is a positive smooth function on [r 1 + , π 2 l]. Moreover, V 1 is a smooth function. Thus, we can apply [83, Chapter 11, Lemma 3.1] and since the interval [r 1 + , π 2 l] is compact, we conclude that (4.33) holds true.

Definition of the non-Diophantine condition as the set P Blow-up
We first define Wronskians of solutions of the radial o.d.e. which will play a fundamental role in the estimates.
Note that this is well-defined as the Wronskians W 1 and W 2 only depend on P (by construction). Moreover, they are manifestly continuous functions on P for fixed m, .
Remark 5.1. Note that, as discussed in the introduction, the Wronskian W 1 does not vanish. Nevertheless, W 1 can be very small (as m, → ∞) which corresponds to frequency parameters associated to stable trapping. On the other hand, W 2 may vanish and this indeed corresponds to stable trapping. In particular, if W 2 vanishes, then the solution u ∞ is a multiple of the u Ai2 which is exponentially damped in the semi-classical forbidden region. In this case, we infer that W 1 is exponentially small and indeed, we are in the situation of stable trapping. This would then show that there exists a quasimode with frequency ω = ω − m This intuition leads to the following non-Diophantine condition for the set P Blow-up . While a priori the set P Blow-up could be empty, we will show in the following that it is dense in P and Baire-generic, i.e. a countable intersection of open and dense sets.

Topological genericity: P Blow-up is Baire-generic
We will first show that each U m0 is dense. To do so, we let m 0 and p 0 = (m 0 , a 0 ) ∈ P be arbitrary and fixed throughout Section 5.2. Also, let U ⊂ P be an open neighborhood of p 0 . We will show that there exists an element of U m0 which is contained in U. We now define a curve of parameters through p 0 as follows.
Proof. By construction of E p in (4.6) and Lemma 5.1, it follows that for r ≥ r 2 . Thus, by choosing δ > 0 sufficiently small andλ 0 ∈ (λ 1 , λ 2 ) \ N p0 , we have that for all i ∈ N, for all parameters in γ δ (p 0 ). Now, recall the definition of W 1 and W 2 from Definition 5.1.
Proposition 5.1. Let m 0 ∈ N. Then, there exist a parameter as well as The proof of Proposition 5.1 relies on the following two lemmata and will be given thereafter. First, we will start by showing that for every m i ≥ m 0 sufficiently large, there exists a p Blow-up ∈ γ δ (p 0 ) such that W 2 = 0 and |ΓW 2 | > 1. We will state this as the following lemma.
Proof. Throughout the proof of Lemma 5.3 we will use the convention that all constants appearing in , , ∼ and O only depend on p 0 , l and δ > 0.
Letm 0 > 0. We begin by showing 1. From Lemma 4.3 and (5.14) we have uniformly on γ δ (p 0 ). Now, for all m i >m 0 sufficiently large, we use the asymptotics for the Airy functions as shown in Lemma A.1 to conclude that Thus, in order to conclude that W[u Ai2 , u ∞ ](m i , ϑ) = 0 for some value on γ δ (p 0 ), we have to vary p(ϑ) ∈ γ δ (p 0 ) such that m i ξ ∞ goes through a period of π. Thus, it suffices to let ξ ∞ go through a period of πm −1 i . From (5.13) we have uniformly on γ δ (p 0 ). Thus, by potentially choosing m i >m 0 even larger, there exists a parameter ϑ Blow-up with Having found m i and ϑ Blow-up , we will now prove 2. We again use Lemma A.1, (4.50) and an analogous computation as in (5.20) to conclude that for ϑ = ϑ Blow-up we have |u Ai2 (r * = lπ/2, m i )| ∼f on γ δ (p 0 ) in view of (5.13). Now, we take the derivative of (5.20) with respect to Γ. Since |l 2 ΓV main | 1 and |l 2 ΓV 1 | 1, we have that ΓO(m  for all m i sufficiently large. Moreover, there exist constants α 1 = α 1 (m i ) ∈ R and β 1 = β 1 (m i ) ∈ R satisfying |α 1 | e −cmi and |β 1 | e −cmi such that u Ai2 = α 1 u Ai1 + β 1 u Bi1 .
Proof. We start by proving |W(u Ai2 , u Bi1 )| f 1 2 1 (r * 1 )e −cmi . Choose > 0 sufficiently small but fixed and evaluate the Wronskian W(u Ai2 , u Bi1 ) at r * := r * 1 + . Then, using standard bounds on Airy functions from Lemma A.1 we obtain from which the first estimate follows by evaluating the Wronskian W(u Ai2 , u Bi1 ) at r * = r * 1 + and the fact thatf 1 (r * 1 )/f 2 (r * 2 ) ∼ 1. The second estimate of (5.29) follows in the same manner but it is easier as u Ai2 is already exponentially small in the region between the turning points r * 1 and r * 2 since For the second part of the lemma we first note that To conclude it suffices to show that In view of the error bounds from (4.45)-(4.48) and the chain rule, we conclude that for all m i sufficiently large. Now, we are in the position to prove Proposition 5.1.
Proof of Proposition 5.1. Let m 0 ∈ N be arbitrary. Using Lemma 5.3, we let m i > m 0 and fix p Blow-up ∈ γ δ (p 0 ) ⊂ U such that W 2 = 0 and |ΓW 2 | > 1 as well as . We moreover have where |α ∞ | ∼ m 5 6 i . Thus, in view of Lemma 5.4 we have for all r * sufficiently small and particularly as r * → −∞. Moreover, as r * → −∞, we have that Thus, by potentially choosing m i even larger (i.e. choosem 0 larger in Lemma 5.3) and noting that Proof. Since p 0 ∈ P and U ⊂ P, U p 0 were arbitrary, Proposition 5.1 shows that for any m 0 ∈ N sufficiently large, the set U m0 as defined in Definition 5.2 is dense in P. Since W 1 , W 2 , σ 1 and σ 2 are continuous, U m0 is manifestly open. Thus, in view of Baire's theorem [2], P Blow-up is Baire-generic and in particular dense.

Metric genericity: P Blow-up is Lebesgue-exceptional and 2-packing dimensional
Proposition 5.3. The set P Blow-up is a Lebesgue null set.

Construction of the initial data
Having constructed the set P Blow-up , we will turn to the problem of showing blow-up. We begin by fixing an arbitrary parameter p = (m, a) ∈ P Blow-up . (6.1) which we keep fixed through the rest of the paper, i.e. throughout Section 6, Section 7, Section 8 and Section 9. This also fixes the mass M = ml/ √ 3 and angular momentum a = al/ √ 3. As stated in the conventions in Section 2.3, all constants appearing in , , ∼ and O will now depend on p as fixed in (6.1) (and on l > 0 as fixed in (2.6)) throughout Section 6, Section 7, Section 8 and Section 9.
By construction of P Blow-up and since p ∈ P Blow-up , there exists an infinite sequence and Without loss of generality we also assume that all m i are taken sufficiently large, i.e. m i ≥ m 0 (6.5) for all i ∈ N and for a sufficiently large m 0 = m 0 (p) ∈ N only depending on the choice of p.
We will now carefully choose initial data for (1.2) with compact support in K, which we define in the following. Lemma 6.1. There exists a compact interval K ⊂ (−∞, π 2 l), an > 0 and a constant c > 0 such that for every i ∈ N, there exists a subinterval Moreover, we choose K such that inf K > 3r + .
is a solution to (2.43), i.e. a solution to there exists a δ > 0 and ar * 2 such that V main < −δ for r * ∈ r * 2 , l π 2 , (6.11) see Lemma 4.2. Without loss of generality we can assume thatr * 2 > r * (r = 3r + ). In particular, for m i sufficiently large (take m 0 (p) possibly larger in (6.5)), we have that for r * ∈ [r * 2 , π 2 l). Now, let K =:= [r * 2 , r * 3 ] ⊂ (r * 2 , π 2 l) be a compact subinterval for r * 3 > r * 2 fixed, e.g. r * 3 = 1 2 (r * 2 + l π 2 ). In the region [r * 2 , π 2 l), the smooth potential V main satisfies V main < −δ, |V main | 1 and |V main | 1. (6.13) Moreover, |V 1 | 1 uniformly in [r * 2 , π 2 l). This allows us to approximate u ω− ∞ via a WKB approximation. First, we introduce the error-control function and note that F ∞ ( π 2 l) = 0. In view of the above bounds on V main and V 1 we obtain Vr * 2 , π 2 l (F ∞ ) 1. main (r * = l π 2 )| satisfies |A| ∼ 1 and u∞ satisfies u∞ π 2 l = 0 as well as Now, since u ω− ∞ oscillates with period proportional to m i in view of (6.16), there exists a compact subinterval We are now in the position to define our initial data which will be supported in the compact set K as defined in Lemma 6.1. We assume without loss of generality that all m i are sufficiently large such that we can apply Lemma 6.1. First, let χ : R → [0, 1] be a smooth bump function satisfying χ = 0 for |x| ≥ 1 and χ = 1 for |x| ≤ 1 2 . Then, for i ∈ N we set Definition 6.1. Let m i , i be as in (6.2). For each i ∈ N, let K i ⊂ K be the associated subinterval as specified in Lemma 6.1 and let χ i defined as in (6.19). Then, we define initial data on Σ 0 as ψ Σ0 = Ψ 0 := 0, (6.20) Having set up the initial data we proceed to Definition 6.2. Throughout the rest of Section 7, Section 8 and Section 9 we define ψ ∈ C ∞ (M Kerr-AdS \ CH) to be the unique smooth solution to (1.2) of the mixed Cauchy-boundary value problem with vanishing data on H L ∪ B H , Dirichlet boundary conditions at infinity and the initial data (Ψ 0 , Ψ 1 ) ∈ C ∞ c (Σ 0 ) posed on Σ 0 specified in Definition 6.1. This is well-posed in view of Theorem 2.
Remark 6.1. By a domain of dependence argument one can also view ψ as arising from smooth and compactly supported initial data posed on a spacelike hypersurface connecting both components of I as depicted in Fig. 2.
Remark 6.2. We note that our initial data are only supported on the positive azimuthal frequencies m = m i . The same will apply to the arising solution ψ.
In the following we define the quantity a H from our initial data. This a H will turn out (at least in a limiting sense) to be the (generalized) Fourier transform of the solution ψ H along the event horizon. Definition 6.3. For the initial data Ψ 0 , Ψ 1 as in Definition 6.1 we define Now, we will show that a H has "peaks" at the interior scattering poles ω = ω − m for infinitely many m. This is a consequence of our careful choice of initial data. We formulate this in Lemma 6.2. For a H as in Definition 6.1 we have for (m i , i ) as in (6.2). Proof.
To conclude we use that from (6.3) we have (6.27) 7 Exterior analysis: From the initial data to the event horizon

Cut-off in time and inhomogeneous equation
We will now consider the ψ as defined in Section 6. The goal of this section is to determine the Fourier transform of ψ along the event horizon. To do so we will first take of a time cut-off of ψ. To do so, we let χ : R → [0, 1] be a smooth and monotone cut-off function with χ(x) = 0 for and note that ψ R is smooth and compactly supported in v and satisfies the inhomogeneous equation Analogously, ψ R satisfies the inhomogeneous equation with As in [54, Section 5.1] we have In view of our coordinates, we also have that for each r > r + , the function ψ R (t, r, θ, φ) is compactly supported in R t with values in C ∞ (S 2 ). This allows us to apply Carter's separation of variables to express ψ R as where for each r > r + , is a Schwartz function on R ω with values in C ∞ (S 2 ). We further decompose F[ψ R ](ω, r, θ, φ) in (generalized) spheroidal harmonics is smooth in ω and r > r + for fixed m and and moreover ψ R ∈ L 2 (R ω × Z m × Z ≥|m| ; C ∞ (r, ∞)) (7.11) in view of Plancherel's theorem for everyr > r + . Equivalently, we havê for each r > r + . Now, note that ψ R (v, r, θ,φ + ) is smooth and compactly supported on R v all the way to r = r + and thus, takes values in the space C ∞ ([r + , ∞) r × S 2 θ,φ+ ). After a change of coordinates in (7.12) we obtain thatψ extends smoothly to r = r + (r * → −∞). Similarly to the above, we have ΣF R (ω, m, , r) = 1 √ 2π R S 2 ΣF R (t, r, θ, φ)e iωt e −imφ S m (aω, cos θ)dσ S 2 dt. (7.14) Now, we define Sinceψ R defined in (7.10) is smooth, we have that u R as defined in (7.15) is a smooth function of ω and r > r + . Moreover, we can also differentiate under the integral sign in (7.12) and since ψ R satisfies (7.3), we have that u R satisfies the inhomogeneous radial o.d.e.
In addition, pointwise as → 0. Moreover, we have Proof. We start with the decomposition We first consider F and write its (generalized) Fourier transform as Recall that for all > 0 sufficiently small, we have that v(t, r, θ, φ) = t on the support of F . Thus, as → 0 in the sense of distributions (compactly supported distributions) on R t with values in C ∞ ((r + , ∞) × S 2 ). Hence, pointwise. Thus, pointwise as → 0. But recall that F is compactly supported in (r + , ∞) for all 0 < < 0 sufficiently small and sup 0< < 0 sup r * | ΣF | < ∞ so we can interchange the integral with the limit → 0. Now, we will show that H R → H as R → ∞. As ψ and its derivatives decay pointwise at a logarithmic rate (see Theorem 2), we obtain sup r∈(r+,∞),θ,φ+∈S 2 as R → ∞. Here, we have used that |u ∞ | ≤ C m ω |r * | which holds true as for each ω, m, there exist constants a s , a c only depending on the ω, m, such that u ∞ = a s u s + a c u c , where u s and u c are solutions to the radial o.d.e. satisfying u s ∼ sin((ω−ω+m)r * ) ω−ω+m and u c ∼ cos((ω − ω + m)r * ) as r * → −∞. In the case ω = ω + m, this reduces to u s ∼ r * and u c ∼ 1 as r * → −∞. Hence, a R H → a H as R → ∞ pointwise for each ω, m, .

Representation formula for ψ at the event horizon
In what follows we will prove a representation formula of the truncated solution ψ R along the event horizon in terms of the initial data and an error term which vanishes in the limit R → ∞. More precisely, we will represent ψ R through a R H which is defined in (7.32). Note that in the limit R → ∞, we have a R H → a H which in turn only depends on the initial data (see Definition 6.3). Proposition 7.1. Let a R H be as defined in (7.32). Then, on the event horizon H R we have ψ R (v, r + , θ,φ + ) = 1 2π(r 2 + + a 2 ) m R a R H S m (aω, cos θ)e imφ+ e −iωv dω, (7.43) in L 2 (R v × S 2 ). Moreover, ψ R (v, r + , θ,φ + )S m (aω, cos θ)e −imφ+ e iωv dσ S 2 dv (7.44) pointwise and in L 2 (R ω × Z m × Z ≥|m| ).
Proof. We have ψ R (v, r, θ,φ + ) = 1 2π(r 2 + a 2 ) m R e i(ω−ω+m)r * u R S m (aω, cos θ)e imφ+ e −iωv dω (7.45) and for r + < r < r + + η. Now, since ψ R is compactly supported in v uniformly as r * → −∞, we can interchange the limit r * → −∞ with the integral over v. Thus, sending r → r + (r * → −∞) in (7.46) yields in view of Lemma 7 where a R ,H is defined in (7.27). Now we will perform the limit → 0 on both sides of (7.47) independently. First, from Lemma 7.3 we have that as → 0 pointwise. Moreover, ψ R has compact support on R v uniformly as → 0 and ψ R → ψ R pointwise and in L 2 (R v × S 2 ) as → 0. Thus, the right hand side of (7.47) converges pointwise and due to Plancherel also in L 2 (R ω × Z m × Z ≥|m| ) as → 0. Hence, a R ,H → a R H also holds in L 2 (R ω × Z m × Z ≥|m| ) and we conclude which holds pointwise and in L 2 (R ω × Z m × Z ≥|m| ). And by Plancherel we also have 8 Interior analysis: Estimates on radial o.d.e. and interior scattering poles Having established the behavior of our solution ψ on the exterior R, we will now consider the interior region B characterized by r ∈ (r − , r + ). We will first consider the interior radial o.d.e. and prove a suitable representation formula on the interior. We also recall that in the interior region the tortoise coordinate is defined in (2.18) as where r * ( r++r− 2 ) = 0 and that ∆ < 0 in the whole interior region.
Remark 8.1. As our initial data are only supported on azimuthal modes m which are large and positive, we only need to consider m sufficiently large.

Radial o.d.e. on the interior: fixed frequency scattering
We recall the radial o.d.e. (2.39) and write it in the interior r − < r < r + as where and V 1 is defined in (2.41). Note that L ≥ 0 follows from [54, Lemma 5.4]. Also note that V 1 = O(|∆|) uniformly for r * ∈ (−∞, ∞). We will treat V 1 as a perturbation and recall that the high-frequency part of the potential is given by ) We now define fundamental pairs of solutions to the radial o.d.e. corresponding to the event and Cauchy horizon, respectively. Definition 8.1. We define solutions u H R , u H L to (8.2) in the interior through the condition for r * → −∞. For ω = ω + m, they form a fundamental pair. For ω = ω + m the solutions u H L and u H R are linearly dependent. Analogously, we define as r * → +∞. For ω = ω − m, they form a fundamental pair. For ω = ω − m the solutions u CH L and u CH R are linearly dependent.
We moreover define reflection and transmission coefficients. Equivalently, we have They satisfy the Wronskian identity Further, we define the renormalized transmission and reflection coefficient We begin by showing L ∞ estimates for the solutions defined in (8.1). To do so we will consider the cases |ω − ω − rm| ≥ cut m for all r ∈ [r − , r + ]. Note that cut > 0 will be fixed in Lemma 8.2 only depending on the black hole parameters. Lemma 8.1. Assume that |ω − ω r m| ≥ cut m for all r ∈ [r − , r + ], for some cut > 0. Then, Proof. We first consider the case that ω − ω r m ≥ cut m for all r ∈ [r − , r + ]. Moreover, from our assumption, we have that the principal part of the potential V satisfies 8.20) and the error term satisfies |V 1 | |∆|. Thus, the error control function m . This allows us to apply standard estimates on WKB approximation such as [83, Chapter 6, Theorem 2.2] and deduce that Similarly, we show that the above holds for ω r m − ω ≥ cut m. This shows (8.17). The bounds (8.18) and (8.19) are shown completely analogously and their proofs are omitted.
There exists a constant cut > 0 such that the following holds true. Assume that |ω − ω r m| ≤ cut m for some r ∈ [r − , r + ], then L m 2 .
Proof. Note that L is larger than the lowest eigenvalue of the operator P (aω) + a 2 ω 2 − 2aΞωm, where P is as in (2.37). Since it suffices to show that the second term is bounded from below by O(m 2 ). To do so, let r ∈ [r − , r + ] such that |ω − ω r m| ≤ cut m. Then, in view of we conclude for sufficiently small cut > 0.

(8.39)
Moreover, Proof. From Lemma 8.2 we know that L m 2 . Now, we write u H R as the solution to the Volterra equation where the kernel is given by For y ∈ (−∞, R 1 ), a direct computation shows for and analogously for u H R B . Evaluating the Wronskians at r * = R 1 , we obtain in view of (8.46) and (8.47). This shows that To show the bound on ∂ ω u H R we considerũ H R . Then,ũ H R satisfies the Volterra equatioñ whereK(r * , y) = e i(ω−ω+m)(r * −y) sin((ω−ω+m)(r * −y)) ω−ω+m . Completely analogous to before, it follows that As |∂ ω λ m (aω)| |m| from Lemma and Evaluating this at r * = R 1 yields and   With the above lemma in hand we conclude Lemma 8.5. Let m ∈ N be sufficiently large and let > 0 be sufficiently small only depending on the black hole parameters. Then, Proof. We again only show the claim forũ H R as the other cases are completely analogous. Assume that |ω − ω + m| ≤ for some > 0 sufficiently small. In view of Lemma 8.3 it suffices to consider the region r * ∈ [R 1 , 0]. Now, note that  |∂ ω u H R | L In view ofũ H R = e i(ω−ω+m)r * u H R and the chain rule, the claim follows.
Proof. Throughout the proof we assume that ω = ω − m. As u CH R = u CH L for ω = ω − m, it suffices to bound the Wronskian |W[u H R , u CH R ]| from below. To do so, let A and B be the unique coefficients satisfying Now, for > 0 to be chosen later, define Now, u CH R − 1 is a solution to the Volterra equation Using bounds on solutions to Volterra integral equations as before (see [83,Chapter 6, §10]), we obtain that for > 0 sufficiently small enough. Thus, Note that (8.34) also holds if we replace R 2 by R 2 for some fixed value of > 0. Thus, we conclude that |B| = |A| 1 which shows This concludes the proof. where u H R is defined in (8.1) and

Scattering poles: Representation formula for ψ on the interior
Moreover, in B we havẽ as well asψ Proof. Note that F H [ψ 0 ](ω, m, ) is rapidly decaying and smooth which follows from the fact that ψ 0 ∈ C ∞ c (H R ). Moreover, using Lemma 8.1 and Lemma 8.3, we have that the right hand side of (8.96) is a smooth solution to (1.2) in the interior region B. Now, note that the right hand side of (8.96) converges to ψ 0 as r → r + for fixed v. Similarly, after a change of coordinates to (u, r, θ,φ + ) we obtainψ converges to zero as r → r + and u fixed in view of the Riemann-Lebesgue lemma. Thus, (8.96) follows from the uniqueness of the characteristic problem.
In order to show (8.98) we first write the right-hand side of (8.96) as a principal value integral and then use the definition of the reflection and transmission coefficients from Definition 8.2 to replace u H R with In order to use linearity of the principal value to writeψ as a sum of two terms as in (8.98), it suffices to show that converges locally uniformly. Note that the other term with t(ω, m, )u CH L replaced by r(ω, m, )u CH R is treated completely analogously. In the following we will be brief because in the proof of Theorem 1, where we have to quantitatively control terms of the form (8.101), we will indeed show stronger estimates and provide more details. First, in view of the facts that F H [ψ 0 ](ω, m, ) is rapidly decaying in ω, m, , that S m (aω, cos θ) L 2 ((−π,π);cos θdθ) = 1, and that we have uniform (polynomial) bounds on |t(ω, m, )| and u CH L L ∞ ([R1,+∞)) (see Lemma 8.1,Lemma 8.3,Lemma 8.4), it suffices to consider frequencies in the range |aω − aω − m| < 1 m . Now, uniformly in |aω − aω − m| < 1 m , we have polynomial bounds in ω, m, on ∂ ω S m (aω, cos θ) L 2 ((−π,π);cos θdθ) , ∂ ω u CH L L ∞ (0,+∞) and |∂ ω t(ω, m, )| as shown in Proposition 3.3, Lemma 8.5 and Lemma 8.4, respectively. Moreover, again using the bound on ∂ ω S m (aω, cos θ) L 2 and the fact that ψ 0 is compactly supported, we also obtain that sup |aω−aω−m|< 1 m |∂ ω F H [ψ 0 ]| is rapidly decaying in m, . This shows that (8.101) converges locally uniformly for r * ≥ 0, v ∈ R with values in L 2 (S 2 ). This shows that, after a change of coordinates, (8.98) and (8.99) hold true pointwise for r ≤ r−+r+ 2 , v ∈ R and in L 2 (S 2 ). Finally, using standard elliptic estimates and the fundamental theorem of calculus, we also have polynomial bounds in ω, m, on S m (aω, cos θ) L ∞ (π,π) , as well as polynomial bounds in m, on sup |aω−aω−m|< 1 m ∂ ω S m (aω, cos θ) L ∞ (−π,π) . Thus, both terms on the right-hand side of (8.98) are continuous and the equality (8.98) holds pointwise. We obtain the analogous result for (8.99).
Before we prove the blow-up result in Section 9, we need one more final ingredient which is a consequence of the domain of dependence. Lemma 8.7. Letψ ∈ C ∞ (B) be a solution to (1.2) arising from vanishing data on H L ∪ B H and compatible smooth data ψ 0 ∈ C ∞ (H R ). Then,ψ(u 0 , r 0 , θ 0 , φ * − ) only depends on ψ 0 {v≤2r * (r0)−u0+c} , wherec =c(p, l) > 0 is a constant.
We will now finally turn to the proof of Theorem 1.

Proof of Theorem 1: Small divisors lead to blow-up
We recall that the cosmological constant Λ < 0 (and thus l = −3/Λ > 0) was arbitrary but fixed as in (2.6).
Theorem 1. Conjecture 5 holds true. More precisely, let the dimensionless black hole parameters (m, a) ∈ P Blow-up be arbitrary but fixed as in (6.1), where P Blow-up is defined in Definition 5.3.
Proof. We decompose the term I b res = I b1 res + I b2 res further into the two summands appearing in the ω-integral. We will estimate each of them individually. We begin with I b1 res and estimate I b1 for all |ξ − ω − m| < 1 m which in turn is a consequence of Lemma 3.2.
Proof. We use that the spheroidal harmonics S m (amω − , cos θ)e imφ * − form an orthonormal basis of L 2 (S 2 ) to estimate Ĩ a res −Î a where we used the Cauchy-Schwarz inequality in the last step. Now, we turn toĨ a res as defined in (9.26) and first only consider the ω-integral √ 2π e 2iω−mr * n iπ sgn ψ n 0 m (· − u 0 + 2r * n )e iω−m· , (9.29) where sgn has to be understood as a Schwartz distribution. We have used that F p.v. 1 ω = iπsgn in the sense of distributions. Now, since ψ 0 is smooth, the function v → ψ n 0 m is a Schwartz function with values in the space of superpolynomially decaying sequences in m, as a subspace of L 2 (Z m × Z ≥|m| ). Particularly, this implies that Completely analogous to the the analysis before, we also decompose II as II = II res + II non-res = II res + II non-res  Hence, using we obtain e imφ * − S m (amω − , cos θ),Ĩ a res +ĨI a res L 2 (S 2 ) = −iπ r 2 + + a 2 r 2 n + a 2 Now, by construction of ψ n 0 , we have that ψ n 0 m (v) = 0 for v ≥ 2r * n − u 0 +c, wherec is a constant only depending on the black hole parameters. In particular, this implies that m iπ r 2 + + a 2 r 2 n + a 2 where |Err(D)| u0 D uniformly for all r * n ≥ r * 0 . We have established formula (9.6) which concludes the proof of Proposition 9.1.
Proof of Corollary 1. LetΨ 0 ,Ψ 1 be initial data as in the statement of Corollary 1 and letψ the arising solution to (1.2). In view of the fact that different azimuthal modes m are L 2 (S 2 )orthogonal in evolution, it suffices to show the blow-up for the modes m = m i , where m i is the sequence in (6.2) associated to the non-Diophantine condition (6.3). Now, the proof of Theorem 1 also carries over for the initial dataΨ 0 ,Ψ 1 and in particularly the analog of (9.44) holds true.