Fast dimension spectrum for a potential with a logarithmic singularity

We regard the classic Thue--Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of the local measure decay (superpolynomial). For several scaling functions, we show that points with this behavior are abundant in the sense of full Hausdorff dimension. At the fastest possible scaling, the corresponding rates reveal several remarkable phenomena. There is a gap between level sets for dyadic rationals and non-dyadic points, and beyond dyadic rationals, non-zero accumulation points occur only within intervals of positive length. The dependence between the smallest and the largest accumulation point also manifests itself in a non-trivial joint dimension spectrum.


Introduction and main results
The study of potential functions ψ over an expanding dynamical system (X, T ) and the corresponding equilibrium measures has a long and rich history; for a few classical references relevant for this work compare [6,18,20].If the potential function ψ is sufficiently regular, the full strength of the thermodynamic formalism is applicable.Using standard results in multifractal analysis, this yields a detailed description of both the Birkhoff averages of the potential function and of the local dimensions of the equilibrium measure.More precisely, one considers and the corresponding dimension spectrum, which is given by the Hausdorff dimension of the corresponding level sets, If ψ is Hölder continuous (and the dynamical system sufficiently nice), the dimension spectrum f ψ is known to be given by a concave real analytic function, supported on a finite interval, outside of which the level sets are empty [18].In this setting, the local dimension d µ (x) = lim r→0 log µ(B r (x)) log(r) of the unique equilibrium measure µ coincides with the Birkhoff average b ψ (x) up to a constant (whenever any of the limits exists).A multifractal analysis of d µ is therefore obtained along the same lines.Over the last decades, similar results have been established under less restrictive regularity assumptions.At the same time, the study of singular (or unbounded) potentials has gained increased attention.In the presence of a singularity, the dimension spectra can be positive on a half-line and the points with infinite Birkhoff averages (or infinite local dimensions of the equilibrium measure) may have full Hausdorff dimension.In this case, a more complete understanding can be obtained by renormalizing the Birkhoff sums (or the measure decay on shrinking balls) with a more quickly increasing function.This was studied for the specific case of the Saint-Petersburg potential in [14] and in the context of continued fraction expansions; see for example [9,15].
In this note, we contribute to the study of singular potentials and their equilibrium measures via a case study of the Thue-Morse (TM) measure.This measure was one of the first examples of a singular continuous measure, exhibited by Mahler almost a century ago [17].To this day, it is of interest in number theory and the study of substitution dynamical systems and continues to be the object of active research-compare the review [19] for a collection of recent results and open questions.It can be written as an infinite Riesz product on the torus T (identified with the unit interval) via to be understood as a weak limit of absolutely continuous probability measures.The TMmeasure falls into the class of g-measures [13], most recently renamed "Doeblin measures" in [4], giving credit to the pioneering role of Doeblin and Fortet [8].This class of measures had an important role in fueling the development of the thermodynamic formalism, largely due to the contributions by Walters [21,22] and Ledrappier [16].The term "g-measure" is related to the observation that µ TM can be constructed by tracing a (normalized) function g, in this case given by along the doubling map T : x → 2x mod 1; see Section 2 for details and a formal definition of the term g-measure in our setting.
The doubling map (T, T ) is closely related to the full shift (X, σ), with X = {0, 1} N and σ(x) n = x n+1 via the (inverse) binary representation π semi-conjugates the action of σ and T .The map π 2 is 2-to-1 on the set D of sequences that are eventually constant (preimages of dyadic rationals), and 1-to-1 everywhere else.Since the dyadic rationals are countable and hence a nullset of µ TM , we can uniquely lift µ TM to a measure µ on X satisfying We adopt a standard choice for the metric on X, given by d(x, y) = 2 −k+1 whenever k is the smallest integer with x k ̸ = y k .We also employ for every finite word w ∈ {0, 1} n and n ∈ N the cylinder set notation [w] = {x ∈ X : The choice to work with (X, σ) instead of (T, T ) is purely conventional and mostly made for the sake of a simpler exposition.All of the results presented in this section hold just the same over the torus and the proof works in the same way with a few minor adaptations.
The close relation between µ and g alluded to earlier, persists in a thermodynamic description of µ.Indeed, due to a classical result by Ledrappier [16], µ can alternatively be characterized as the unique equilibrium measure of the potential function which has a singularity at the preimages of the origin, x = 0 ∞ and x = 1 ∞ .A multifractal analysis for the Birkhoff averages b ψ and the local dimensions d µ was performed in [1,10].There it was shown in particular that the level sets have full Hausdorff dimension as soon as α ⩾ 2. This supports the idea that a superpolynomial scaling of the the TM measure (and a superlinear growth of the Birkhoff sums) is in some sense typical for the TM measure.We pursue this idea in the following.Since the ball of radius 2 −n around x ∈ X is given by we may also write the local dimension of the measure µ as −n log 2 , provided that the limit exists.The equilibrium state can be expected to avoid the singularities at the preimages of the origin (which are also fixed points of the dynamics).It is therefore reasonable to expect the fastest possible decay rate for µ at these positions.Given π 2 (x) = 0, it was already observed in [11] (for more refined estimates see also [2,3]) that The same conclusion holds in fact for x ∈ D, the preimages of dyadic rationals [12] (and no other points, as we will see below).However, this is a countable set of vanishing Hausdorff dimension.It seems natural to inquire if sets of non-trivial Hausdorff dimension occur if n 2 is replaced by a different scaling function.
When it comes to the Birkhoff sums, choosing x ∈ D immediately gives S n ψ(x) = −∞ for large enough n, so we will not get a finite result for any scaling function.However, as long as x / ∈ D, we will obtain and in this sense the fastest possible scaling for S n ψ is also given by n 2 .We may interpolate between the linear and quadratic scaling via the scaling function n γ for some γ ∈ (1, 2).It turns out that the points with such an intermediate scaling have full Hausdorff dimension.
Theorem 1.1.For each γ ∈ (1, 2) and α ⩾ 0, the level sets In this sense, n 2 is the critical scaling, at least for phenomena that can be distinguished via Hausdorff dimension.We will therefore focus on accumulation points for this particular scaling in the following.
Although the relation between S n ψ(x) and µ(C n (x)) is not as simple as in the Hölder continuous case, their asymptotic behavior is still closely related.In fact, both expressions can be controlled via an appropriate recoding of x ∈ X.As long as x / ∈ D, its binary representation can be uniquely written in an alternating form as x = a n 1 b n 2 a n 3 b n 4 . . ., where a, b ∈ {0, 1} with a ̸ = b and n i ∈ N for all i ∈ N.With this notation, the alternation coding is a map τ : X \ D → N N , given by for all m ∈ N.For notational convenience, we also set F (x) = lim sup m→∞ F m (x) and F (x) = lim inf m→∞ F m (x).The role of this sequence of functions is clarified by the following result.
This has the following remarkable consequence.
Corollary 1.3.Whenever the sequence log µ(C n (x))/n 2 has a non-trivial accumulation point (̸ = 0), the accumulation points form in fact an interval of strictly positive length.The same conclusion holds for the sequence S n ψ(x)/n 2 .
Also, we immediately obtain a gap result for dyadic vs non-dyadic points.
Corollary 1.4.If x ∈ D, then S n ψ(x) = −∞ for large enough n, and Due to the pointwise relation in Proposition 1.2, it suffices to focus on the accumulation points of (F m ) m∈N .These can be analysed via the joint (dimension) spectrum of F and F , given by (α, β) → dim H {x : More generally, we calculate the Hausdorff dimension of for every subset S ∈ R 2 .Since all accumulation points of (F m ) m∈N are in [0, 1], the pair (F , F ) is certainly contained in It therefore suffices to consider sets S ⊂ ∆.We show that the joint spectrum is given by a function f : ∆ → [0, 1], defined on ∆ \ {(0, 0)} as see Figure 1 for an illustration.A continuous extension of f to ∆ is not possible, since f can take arbitrary values in [0, 1] as we approach the origin from different directions.We define f (0, 0) := 1, which is the most adequate choice for our application below.
Because of its central role, we detail some properties of the function f below (without proof), which may be verified using standard tools from analysis.We describe the values of f on the boundary of ∆ in the first two items and proceed to monotonicity properties thereafter.(5) For every α ∈ (0, 1), there is a value is strictly increasing on (α, α * ), takes its maximum in β = α * and is strictly decreasing on (α * , 1).
Especially the last property in Proposition 1.6 is remarkable as it shows that, for a fixed value F ∈ (0, 1), most points (in the sense of Hausdorff dimension) achieve a value of F that lies strictly between F and 1.We emphasize that, due to Proposition 1.2, the result in Theorem 1.5 can also be regarded as a statement about the level sets for the lim inf and lim sup of the sequences log µ(C n (x)/n 2 and S n ψ(x)/n 2 , respectively.In particular, the non-triviality of the joint spectrum of the lim sup and the lim inf persists.Let us single out two more consequences for the reader's convenience.

Estimates for Birkhoff sums and measure decay
We begin with a few preliminaries on notation and basic concepts.Given two (realvalued) sequences (f m ) m∈N and (g m ) m∈N , we write Every Borel probability measure ν on X may also be regarded as a linear functional on the space of continuous functions C(X).This motivates the notation ν(f ) := f dν for f ∈ C(X), which we sometimes extend to ν-integrable functions f .Following [13,16], a g-function over (X, σ) is a Borel measurable function g : X → [0, 1] satisfying y∈σ −1 x g(y) = 1 for all x ∈ X.There is a corresponding transfer operator We call ν a g-measure with respect to indeed a g-function with g-measure µ; compare [2] for the corresponding statement about g and µ TM over the doubling map.In fact µ TM is known to be the unique g-measure with respect to g.We refer to [4,5,7,13] and the references therein for more on the (non-)uniqueness of g-measures.
Since g = exp •ψ, the invariance of µ under L g builds a natural bridge to the potential function.This can be used to obtain the following replacement for the Gibbs property in the Hölder continuous case.
Lemma 2.1.For any two words w ∈ {0, 1} n and v ∈ {0, 1} m , we have g n (wx) dµ(x), where In particular, Proof.Writing 1 [wv] for the characteristic function of [wv] and using the invariance of µ under the transfer operator, we get and obtain via a straightforward calculation This yields the first assertion.The inequalities follow by estimating the integrand via its infimum (or supremum) and taking the logarithm.□ We continue by recording a basic estimate for the potential function.The proof is straightforward and left to the interested reader.
Lemma 2.2.For every x ∈ T, let |x| be the smallest Euclidean distance to an endpoint of the unit interval.Then, we have We use these bounds to obtain an estimate for S n ψ(x) for arbitrary n ∈ N and x ∈ X\D.
In the special case k ′ = k, this yields By symmetry, the same bounds hold if y ∈ [1 k 0].For simplicity let us assume that All other cases work analogously.Since n + r m+1 = N m+1 = N m + n m+1 , we have in particular that n − N m = n m+1 − r m+1 .Using this, we can split up the Birkhoff sum as using ( 2) and (3) in the last step.This shows the lower bound.The upper bound follows along the same lines.□ Although µ(C n (x)) is closely related to S n ψ(x) via Lemma 2.1, we emphasize that, in contrast to S n ψ(x), the expression µ(C n (x)) depends only on the first n positions of x.To account for this fact, we extend the action of the alternation coding τ to finite words via for a ̸ = b, (and m odd) and accordingly if the word ends in b nm (if m is even).
Lemma 2.4.Let w ∈ {0, 1} n with τ (w) = (n 1 , . . ., n m ).Then, Proof.Again, it suffices to consider the case that w is of the form From Lemma 2.1 (and using µ[0] = 1/2 by symmetry considerations), we obtain inf For the lower bound, we can apply Lemma 2.3 to x ∈ [w0] with n = N m and r m+1 = n m+1 which immediately gives the desired estimate.For the upper bound, assume that x ∈ [w] and note that in this case, τ (x) is of the form and may argue as for the lower bound.We hence assume N m−1 < n < N m in the following.Then, r m = N m − n is equal to n m (x) − n m by construction.From this, we easily conclude that Combining this estimate with the upper bound provided by Lemma 2.3 yields Since x ∈ [w] was arbitrary, this concludes the proof via (4).□ We summarize our findings in terms of the function sequence (f m ) m∈N , with for all x ∈ X \ D and m ∈ N.For an illustration of the following proposition we refer to Figure 2.

Intermediate scaling
In this Section we investigate the scaling function n → n γ for γ ∈ (1, 2) and prove that this scaling is typical for S n ψ(x) and log µ(C n (x)) in the sense of full Hausdorff dimension.As a first step, we show that we may restrict our attention to the limiting behavior of f m as m → ∞.
Proof.First, we will show that the convergence of N −γ m f m (x) implies that both n m /N m and n 2 m /N γ m converge to 0. Indeed, whenever n m /N m > δ > 0, we get f m (x) ⩾ δ 2 N 2 m , which can happen only for finitely many values of m.This implies also lim m→∞ N m /N m+1 = 1.Finally, if n 2 m /N γ m > δ > 0 for infinitely many m, we obtain and applying the lim sup to both sides yields α ⩾ α+δ, a contradiction.These observations offer enough control over the points N m ⩽ n < N m+1 to obtain the desired convergence from Proposition 2.5 (and the fact that 0 ⩽ r m , s m ⩽ n m in the corresponding notation).□ In order to establish lower bounds for the Hausdorff dimension of level sets, we will make use of the following simple consequence of the mass distribution principle.Recall that we define the upper density of a subset M ⊂ N via Proof.We define a Bernoulli-like measure ν on A by "ignoring the determined positions".More precisely, for every n ∈ N let P n = {1, . . ., n} \ M be the free positions and set c n = #P n .Clearly, there are 2 cn choices for v ∈ {0, 1} n such that [v] intersects A and we set otherwise.It is straightforward to check that this definition is consistent and there is a unique measure ν with this property by the Kolmogorov extension theorem.We obtain for every x ∈ A and n ∈ N that ν(C n (x)) = 2 −cn and therefore the lower local dimension of ν at x is given by The claim hence follows via the (non-uniform) mass distribution principle.□ With the help of Lemma 3.2, we will show that for every β > 0, the situation in Lemma 3.1 is typical in the sense of full Hausdorff dimension.Proposition 3.3.For every γ ∈ (1, 2) and α > 0, we have We construct a subset with Hausdorff dimension arbitrarily close to 1.The dimension estimate will be provided by Lemma 3.2.Hence, we want to find a subset M ∈ N of arbitrarily small upper density, such that fixing x on M in an appropriate way ensures that f m (x) ∼ αN γ m .The general strategy is the following: We choose a sequence (θ k ) k∈N of positive real numbers such that θ m , we fix x ∈ X to be constant on an interval of some appropriate length c k in [θ k , θ k+1 ], and to have bounded alternation blocks outside of these intervals.Using that c k grows slower than θ k+1 − θ k , this will fix x on a set of positions with arbitrarily small density.The details follow.
For definiteness, we fix some large number r = r(γ) (the exact value will be determined later) and set θ k = k r .For r > 1 this satisfies θ k − θ k−1 → ∞ and θ k−1 /θ k → 1 for k → ∞, as required.An appropriate choice of c k turns out to be For this to grow slower than θ k − θ k−1 , we require that δ < r − 1.Since this holds true for large enough r and we take some r = r(γ) > 2 with this property.Hence, we can choose k 0 ∈ N such that c k < θ k − θ k−1 for all k ⩾ k 0 .We specify a set of positions via and define To avoid large blocks outside of M 1 we further fix a large cutoff-value Λ ∈ N and set

Finally, we combine both conditions by setting
is the alternation coding of x, this implies that for every k there is a unique index i k such that and in particular n i k ⩾ c k − 2. Since R Λ restricts the length of blocks outside of M 1 , we find that n i k can in fact not be much larger and hence where the implied constant depends on Λ.For all other indices i we have that n i ⩽ Λ is bounded by a constant.Hence, for i k ⩽ m < i k+1 we obtain With the specific choice of c k in (5), we obtain using an integral estimate in the last equation.Since θ k ∼ θ k+1 and by the monotonicity of N m , we also observe that N m ∼ θ k = k r for i k ⩽ m < i k+1 , and therefore as required.That is, A Λ ⊂ {f m ∼ αN γ m } for every Λ ∈ N and it suffices to find an appropriate lower bound for the Hausdorff dimension of A Λ .Since the positions in M 1 are accumulated to the left of the values θ k , we obtain using that δ + 1 < rγ in the last step.Because the points in A Λ are fixed on the positions given by M 1 ∪ ΛN ∪ (ΛN + 1), we obtain by Lemma 3.2, Since this is a lower bound for dim H {f m ∼ αN γ m } and Λ ∈ N was arbitrary, the claim follows. □ Proof of Theorem 1.1.For α > 0, the desired relation follows by combining Proposition 3.3 with Lemma 3.1.For α = 0, simply recall that both S n ψ(x) and log µ(C n (x)) scale linearly with n for a set of full Hausdorff dimension [1].□

Spreading of accumulation points
We specialize to the scaling function n → n 2 for the remainder of this article.We continue with the standing assumption that x ∈ X \ D. By Proposition 2.5, the accumulation points for − log µ(C n (x))/(n 2 log 2) are the same as those of Similarly, the accumulation points of −S n ψ(x)/(n 2 log 2) coincide with those of Recall the notation F m (x) = N −2 m f m (x), together with F (x) = lim inf m→∞ F m (x) and F (x) = lim sup m→∞ F m (x).The strict convexity of the function s → s 2 causes the sequence ξ µ n (x) to take its minimum on [N m , N m+1 ] at some intermediate point, provided that n m+1 is sufficiently large; compare Figure 2.This gives rise to a drop of the lim inf, as compared to F (x).
Proof.We start with the assertion about the lim inf.Let m ∈ N and assume that n = (1 + c)N m (not necessarily n < N m+1 ) for some c ⩾ 0. We obtain with equality if and only if n ⩽ N m+1 .Hence, again with equality precisely if n ⩽ N m+1 .For r > 0, the function is strictly decreasing on [0, r), takes a minimum in c = r and is increasing for c ⩾ r.This yields for and in particular, On the other hand, let and note that if F m k (x) converges to α, then so does c m k as k → ∞.In particular, we find for and the claim on the lim inf follows.
For n = (1 + c)N m let I be the interval of values c such that N m ⩽ n ⩽ N m+1 .Due to the monotonicity properties of c → ϕ F m (x) (c), its maximum on I is obtained on a boundary point.By (6), we hence conclude that ξ µ n (x) ⩽ F m (x) or ξ µ n (x) ⩽ F m+1 (x), with equality if n = N m or n = N m+1 , respectively.This implies the assertion about the lim sup.□ Lemma 4.2.Given F (x) = α and F (x) = β, we have Proof.This is similar to the proof of Lemma 4.1.With m ∈ N and n = (1 − c)N m for some 0 ⩽ c < 1, we get with equality if and only if n ⩾ N m−1 .Therefore, again with equality if and only if n ⩾ N m−1 .For 0 < r < 1, the function φr (c) is strictly increasing on [0, r), takes a maximum in c = r and is decreasing for c ∈ (r, 1).Noting that φr (r) = r/(1 − r), the rest follows precisely as in the proof of Lemma 4.1.□ Proof of Proposition 1.2.The corresponding statements for the accumulation points of (ξ µ n (x)) n∈N and (ξ ψ n (x)) n∈N are given in Lemma 4.1 and Lemma 4.2.Combining this with Proposition 2.5 gives the desired relations for the Birkhoff sums and the measure decay.□

Lower bounds
We want to establish necessary and sufficient criteria for x to satisfy F (x) = α and F (x) = β.We show that this requires a certain number of large blocks in the alternation coding τ (x) = (n i ) i∈N .To be more precise, let us start with a certain large cutoff-value Λ ∈ N and let It will be convenient to ignore all contributions of n i to F n (x) as long as n i < Λ.This is achieved by setting Hence, (F Λ m (x)) m∈N and (F m (x)) m∈N have the same set of accumulation points.
Proof.This follows by which gives the desired estimate.□ In principle, it is possible that F Λ m (x) = 0 for all m ∈ N.However, this can only happen if F (x) = 0, a case that we will treat separately.In the following, we always assume that m is large enough to ensure F Λ m (x) > 0.
Example for the alternation block decomposition of x, given that All blocks between N r and N k−1 have length below Λ.
If n j+1 < Λ, we interpolate F Λ r (x) continuously between r = j and r = j + 1 by setting N r = N j + (r − j)n j+1 and For a lower bound on the Hausdorff dimension of {F = α, F = β}, we wish to provide a mechanism that produces an abundance of points with this property.More precisely, we exhibit a subset of {F = α, F = β} that permits a lower estimate for its the Hausdorff dimension via Lemma 3.2.The main idea is the following: Given r ∈ R with F Λ r (x) = β we introduce blocks of length smaller than Λ until we hit the level F Λ k−1 (x) = α for some k ∈ N. Since these blocks can be chosen arbitrarily we interpret them as degrees of freedom or "undetermined positions".We then add a single large block of size n k (the "determined positions") that raises the level back to F Λ k (x) = β; compare Figure 3 for an illustration.The relative amount f (α, β) of undetermined positions turns out to be independent of the starting position r.Repeating this procedure, the lower density of undetermined positions equals f (α, β) over the whole sequence.This will yield the same value as a lower bound on the Hausdorff dimension of {F = α, F = β}.In Section 6, we will prove that this strategy is indeed optimal, establishing f (α, β) also as an upper bound for the Hausdorff dimension.Lemma 5.2.Let j, k ∈ N with j < k and assume that n i < Λ for all j < i < k and n k ⩾ Λ. Suppose that there is j ⩽ r < j + 1 such that Solving for ℓ, we obtain On the other hand, we have n k = mN r by definition, yielding That is, which gives after a few steps of calculation, Finally, this implies .
A few formal manipulations show that this is precisely the expression given by f (α, β).□ order to show that the strategy sketched before Lemma 5.2 is in a certain sense optimal, we move away from the assumption that there are only negligible blocks between N r and N k−1 .In this more general setting, we find the following analogue of Lemma 5.2 which will be useful in Section 6.
Lemma 5.3.Suppose n k ⩾ Λ for some k ∈ N and let Sketch of proof.The proof of Lemma 5.2 carries over verbatim if we replace N r by the term α/βN k−1 and use the identification We can now provide a lower estimate for the dimension of the set {F = α, F = β}.As in Section 3, we will make use of Lemma 3.2, by fixing the values of the sequence x = (x n ) n∈N on an appropriate subset of N. Proof.For β = 0, note that whenever the size of alternation blocks in τ (x) is uniformly bounded, it follows that F (x) = 0. Since the union of all such elements x has full Hausdorff dimension, the claim holds in this particular case.Likewise, the claim is trivial if β = 1 or α = β ̸ = 0 because this implies f (α, β) = 0. We can hence assume α < β < 1 in the following.For simplicity, we further restrict to the case that α > 0. The case α = 0 can be treated by replacing α with a sequence α k → 0 in the argument below.
We follow the ideas outlined before Lemma 5.2, using some of the notation introduced in its proof.For Λ ∈ N, we specify a set of positions, given by and θ 0 ∈ N is a value with mθ 0 > Λ + 2. We recall from the proof of Lemma 5.2 that The set M will denote those positions where the binary expansion of x is assumed to have a (large) constant block.We hence define To avoid contributions that come from the complement of M Λ , we introduce the set Combining both conditions, it is natural to define First, we will show that A Λ ⊂ {F = α, F = β} such that it suffices to bound the Hausdorff dimension of A Λ from below.Let x ∈ A Λ with alternation coding τ (x) = (n i ) i∈N .Since the expansion of x is constant on [(1 + ℓ)θ k , θ k+1 ], there exists a corresponding index and by the assumption on θ 0 this also implies n i k > Λ.On the other hand, the restriction via R Λ ensures that n i k cannot be much larger.More precisely, we have n i k ⩽ mθ k + 2Λ and hence for every k ∈ N. Note that for all other indices i ∈ N the defining condition for R Λ also enforces n i < Λ.Hence, we have I Λ = {i k : k ∈ N 0 } and obtain Clearly, this sequence attains its lim sup along the subsequence with j = i k and k ∈ N.
Since N i k ∼ θ k+1 , we obtain using the first identity from (7) in the last step.On the other hand, (8) implies that the lim inf for F j (x) is obtained along the subsequence with j = i k − 1 and k ∈ N. Since N i k −1 ∼ (1 + ℓ)θ k , we get by a similar calculation as before using the definition of ℓ in the last step.This completes the proof for the statement that In view of Lemma 3.2, one has to compute the upper density of M Λ in order to acquire a lower bound for the Hausdorff dimension of A Λ .Since the elements of M Λ are accumulated to the left of the positions θ k , we have that, where we have used the second identity from (7) in the last step.Since the points in A Λ are determined precisely for the positions in M Λ ∪ ΛN ∪ (ΛN + 1), we get by Lemma 3.2, Proof.For (α, β) ∈ S, we have {(F , F ) ∈ S} ⊃ {F = α, F = β} and we hence obtain dim H {(F , F ) ∈ S} ⩾ f (α, β), due to Proposition 5.4.Taking the supremum over S yields the assertion.□

Upper bounds
We proceed by establishing an upper bound for the Hausdorff dimension of the set {F = α, F = β}.This is somewhat more involved than proving the lower bound because we now have to account for all mechanisms that lead to this particular range of accumulation points.
Let us fix x ∈ X \ D with τ (x) = (n i ) i∈N and Λ ∈ N as in the last section.For every k ∈ N, we define This corresponds to the relative density of positions of x (in the region [1, N k ]) that are occupied with large blocks.Naturally, if we enlarge [1, N k ] by an interval that does not contain elements of I Λ , this density decays.It will be useful to cancel this effect in an appropriate way.To that end, we define a sequence (ϱ k ) k∈N , implicitly dependent on (x, Λ), via , which we may interpret as a renormalized block density.Indeed, one easily verifies that whenever n i < Λ for all j < i ⩽ k, it follows that ϱ j = ϱ k .
In the following, let In the situation of Lemma 5.2, this may be interpreted as the relative size of the single large block n k in the region between N r and N k , normalized by √ β.The similarity of this interpretation with the definition of ϱ k provides some intuition for the following result.Lemma 6.1.Whenever n k ⩾ Λ and where Proof.First, we write ℓ k as a convex combination via Dividing this relation by √ β yields Using n k = N k − N k−1 , the last summand may be rewritten as Note that γ = F Λ k−1 (x) < F Λ k (x) = δ requires that p k ⩽ N k−1 /N k is bounded above by some constant c(δ) < 1, compare the proof of Lemma 4.1.Restricting to those k such that δ > β/2 > 0, we can further assume that there is a uniform p < 1 with c(δ) < p and hence Since ϱ k is non-decreasing we have overall that the distance of ϱ k to η(α ε , β ε ) is nonincreasing and exponentially decaying on a subsequence due to (9).It thereby follows that lim k→∞ ϱ k = η(α ε , β ε ).Hence, we have in every case Proof.Since f (0, 0) = 1 by convention, the lower bound is trivial if (0, 0) ∈ S. We can hence restrict to the case S ⊂ ∆ \ {(0, 0)}.Let x be such that F (x) = α and F (x) = β with (α, β) ∈ S. Take an increasing subsequence (k m ) m∈N such that lim m→∞ F Λ km (x) = β.Then, by Proposition 6.Since Λ ∈ N was arbitrary, this is the desired statement.□ We proceed with two results that provide an estimate for the Hausdorff dimension of level sets of the density function D Λ .First, we recall a standard estimate, including a short proof for the reader's convenience.Note that for every i ∈ I Λ , the number k i of words w r that are completely contained in the corresponding block of length n i satisfies Since n i ⩾ Λ = mk, we can choose k large enough to ensure Hence, using (10), the number r * j is bounded below via As a result we get log ν(C N j (x)) Since ε > 0 was arbitrary, it follows that

Lemma 6 . 5 .
Let ν be a probability measure on X and c > 0. Then,dim H {x ∈ X : d ν x ⩽ c} ⩽ c.Proof.Let A(c) = {x ∈ X : d ν (x) ⩽ c}.Given ε > 0 and n 0 ∈ N, choose for each x ∈ A(c) a number n = n(x) ⩾ n 0 such that the cylinder C n (x) of diameter r n = 2 −n satisfies ν(C n (x)) ⩾ r c+ε n .

d
ν (x) ⩽ lim inf j→∞ log ν(C N j (x)) −N j log 2 ⩽ c log(2 m − 2) m log 2 − log p m log 2 =: c m .Since this holds for all points in B(c) it follows by Lemma 6.5 that dim H B(c) ⩽ c m m→∞ −−−→ c, which indeed implies that dim H B(c) ⩽ c. □ Corollary 6.7.For S ⊂ ∆, we have dim H (F , F ) ∈ S ⩽ sup S f (α, β).Proof.Let c = sup S f (α, β).Due to Proposition 6.4, we have D Λ (x) ⩾ 1 − c for all x ∈ (F , F ) ∈ S and Λ ∈ N.That is, (F , F ) ∈ S ⊂ B(c) in the notation of Lemma 6.6, implying that dim H (F , F ) ∈ S ⩽ dim H B(c) ⩽ c. □ Proof of Theorem 1.5.The lower bound in Theorem 1.5 is given in Corollary 5.5 and the upper bound is provided by Corollary 6.7.□ Achknowledgements