Nodal intersections for arithmetic random waves against a surface

Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus (`3d arithmetic random waves'), we investigate the $1$-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral surface (the `nodal intersection length'). The expected length is proportional to the square root of the eigenvalue, times the surface area, independent of the geometry. Our main finding is the leading asymptotic of the nodal intersection length variance, against a surface of nonvanishing Gauss-Kronecker curvature. The problem is closely related to the theory of lattice points on spheres: by the equidistribution of the lattice points, the variance asymptotic depends only on the geometry of the surface.


Nodal intersections for Laplace eigenfunctions
The nodal set of a function is its zero-locus. Several recent works (e.g. [40,10,18]) studied the number of intersections between the nodal lines of eigenfunctions and a fixed reference curve (nodal intersections on 'generic' surfaces). It is expected that in many situations the nodal intersections number obeys the bound ≪ √ E, where E > 0 is the eigenvalue.
On the three-dimensional standard flat torus T 3 = R 3 /Z 3 , the non-zero Laplace eigenvalues, or energy levels, are of the form 4π 2 m, m ∈ S (3) , where S (3) := {0 < m : m = a 2 1 + a 2 2 + a 2 3 , a i ∈ Z}. Let E = E m := {µ = (µ (1) , µ (2) , µ (3) ) ∈ Z 3 : (µ (1) ) 2 + (µ (2) ) 2 + (µ (3) ) 2 = m} (1.1) be the set of all lattice points on the sphere of radius √ m. The (complexvalued) Laplace eigenfunctions may be written as [5] G with c µ Fourier coefficients. A natural number m is representable as a sum of three integer squares if and only if m = 4 l (8k + 7), for k, l non-negative integers [23,14]. The lattice points number, or equivalently the number of ways to express m as a sum of three squares will be denoted (see e.g. [23, §20]), hence it suffices to consider energies 4π 2 m for m taken up to multiples of 4 (see section 4 for details). It is known [11,35] that the nodal set is a union of smooth surfaces, possibly together with a set of lower dimension (curves and points). Let Σ ⊂ T 3 be a fixed smooth reference surface. Consider the intersection A G ∩ Σ, in the limit m → ∞. Bourgain and Rudnick [5,6] found that, for Σ realanalytic, with nowhere zero Gauss-Kronecker curvature, there exists m Σ such that for every m ≥ m Σ , the following hold: • the surface Σ is not contained in the nodal set of any eigenfunction G m [5, Theorem 1.2]; • the 1-dimensional Hausdorff measure of the intersection has the upper bound for some constant C Σ [6, Theorem 1.1]; • for every eigenfunction G m ,
Definition 1.1. Let Σ be a fixed compact regular 2 toral surface, of (finite) area A. Assume that Σ admits a smooth normal vector locally. We define the nodal intersection length for arithmetic random waves L = L m := h 1 (A Fm ∩ Σ), (1.5) where h 1 is 1-dimensional Hausdorff measure.
We will study L in the high energy limit m → ∞. 1 Defined on some probability space (Ω, F, P), where E denotes the expectation with respect to P. 2 i.e. a smooth 2-dimensional submanifold of T 3 , possibly with boundary. For background on regular surfaces, see e.g. [15,38].  We may view the nodal intersection length L m as another three-dimensional analogue of Z (2) m : indeed, in both cases one considers nodal intersections against a linear manifold.

Prior work on nodal intersections for random waves
The expected number of nodal intersections against smooth curves C of length L is [36,37] This asymptotic behaviour is non-universal: B C (E) depends both on C and on the angular distribution of the lattice points on √ mS 1 , as m → ∞. A nice consequence of (1.8) and (1.9) is that the distribution of Z is asymptotically concentrated at the mean value, i.e., for all ǫ > 0, The leading coefficient in (1.9) is always non-negative and bounded [36, sections 1 and 7]: though it might vanish, for instance when C is a circle, independent of E. Rossi-Wigman [34] investigated the scenario of curves such that 4B C (E)− L 2 vanishes universally ('static curves'). For a density one sequence of energies, the precise asymptotics of the variance in the case of static curves are [34,Theorem 1.3]

Statement of main results
We now state our theorems on expectation and variance of the nodal intersection length (1.5). (1.12) The proof of Proposition 1.2 will be given in section 2. Note that (1.12) is consistent with (1.3), and that (1.12) and (1.8) are of similar shape. In the statement of our next result, − → n (σ) is the unit normal vector to Σ at the point σ. (1.13) Then we have as m → ∞, m ≡ 0, 4, 7 (mod 8), (1.14) Theorem 1.3 will be proven in section 5. Compare the expressions (1.10) and (1.13): while the integral B C (E) depends on both the curve C ⊂ T 2 and on the angular distribution of lattice points on circles, the integral I depends on Σ ⊂ T 3 only. This is because lattice points on spheres are equidistributed 3 (Linnik's problem, see section 4). Our next result concerns the analysis of the quantity I. The integral I satisfies the sharp bounds 15) so that the leading coefficient of (1.14) is always non-negative and bounded: Proposition 1.4 will be proven in section 7. A computation shows that when Σ is a sphere or a hemisphere, the lower bound in (1.15) is achieved, hence the leading term in (1.14) vanishes: in this case the variance is of lower order than m/N (see section 7 for details). As in the problem of nodal intersections against a curve on T 2 [36], the theoretical maximum of the variance leading term is achieved in the case of intersection with a 3 Lattice points on circles equidistribute [19,20] for a density one sequence of energies. To the other extreme, Cilleruelo proved that there exist sequences s.t. all the lattice points lie on arbitrarily short arcs [12,Theorem 2]. The non-uniform densities that emerge ("attainable measures") were partially classified in [26,27]. manifold of identically zero curvature (straight lines in dimension 2, planes in dimension 3). As the case of Σ contained in a plane is excluded by the assumptions of Theorem 1.3, the upper bound of A 2 · π 2 /30 for the leading coefficient in (1.14) is a supremum rather than a maximum, as in [36] (see section 7).
Similarly to [36,37], the above results on expectation and variance have the following consequence.

Outline of the proofs and plan of the paper
Throughout we apply many ideas of [35,26,36,37]. The arithmetic random wave F (d) : T d → R (1.6) is a random field. The number of nodal intersections Z (d) (1.7) against a curve are the zeros of a process, which is the restriction of the random wave F to the curve [36,34,37]. For a smooth process p : T → R defined on an appropriate parameter set T ⊂ R, moments of the number of zeros may be computed, under certain assumptions, via Kac-Rice formulas [2,13].
More generally, given a smooth random field P : T ⊂ open R n → R n ′ , let V be the Hausdorff measure of its zero set. When n−n ′ = 0, V is the number of zeros; when n − n ′ = 1, V is the nodal length of P ; when n − n ′ = 2, V is the nodal area, and so forth. Only the case n ≥ n ′ is interesting, since otherwise the zero set of P is almost surely 4 empty. One may compute, under appropriate assumptions, the moments of V by means of Kac-Rice formulas [2, Theorems 6.2, 6.3, 6.8 and 6.9]. The latter formulas, however, are not applicable to our problem, as the following example illustrates. Example 1.6. Assume that the surface Σ ⊂ T 3 is the graph of a differentiable function, in the sense that it admits everywhere the parametrisation with h ∈ C 2 (U ). We restrict F to Σ, and obtain the random field f : The zero line of f is not necessarily (isometric to) the nodal intersection curve {x ∈ T 3 : F (x) = 0}: rather, it is isometric to the projection of the nodal intersection curve onto the domain U of the parametrisation γ. Therefore, the application of Kac-Rice formulas for f yields the moments of the length of the projected curve (see [2,Theorem 11.3]), not of the intersection curve itself: this is in marked contrast with what happens in the case of the nodal intersections number [36,34,37].
Our approach to the problem begins with the derivation of Kac-Rice formulas for a random field defined on a surface, which is done in section 2 (also see [28,Theorem 5.3] and [29,Theorems 4.1 and 4.4]). Applying the Kac-Rice formula for the expectation (Proposition 2.4 to follow), we will prove Proposition 1.2.
For the nodal intersection length variance we apply Proposition 2.7 and subsequently reduce our problem to estimating the second moment of the covariance function r(σ, σ ′ ) := E[F (σ)F (σ ′ )] (1.17) and of its first and second order derivatives. We thus develop an approximate Kac-Rice formula. To state it, we need some extra notation: first, M := 4π 2 m/3. Moreover, X, X ′ , Y, Y ′ (σ, σ ′ ) are appropriate 2 × 2 matrices, depending on r, its derivatives, and on the surface Σ (see Definition 3.3).
Proposition 1.7 (Approximate Kac-Rice formula). Let Σ ⊂ T 3 be a surface as in Definition 1.1, with nowhere vanishing Gauss-Kronecker curvature. Then we have The proof of this result takes up the whole of section 3. Our problem of computing the nodal intersection length variance (1.14) is thus reduced to estimating the second moment of the covariance function r (1.17) and of its various first and second order derivatives, which is carried out in section 5. The error term in Proposition 1.7 comes from bounding the fourth moment of r and of its derivatives: this is done in section 6. To study the second and fourth moments of r, one needs to understand various properties of the lattice point set E (1.1), covered in section 4. In section 7, we study the leading term of the nodal intersection length variance (1.14), and establish Proposition 1.4. Appendix A is dedicated to proving several auxiliary lemmas.

Future directions
As discussed in section 1.3, for the problem of nodal intersections against a curve in two dimensions, Rossi-Wigman [34] investigated the case of static curves, and found the precise asymptotic behaviour of the variance, for a density one sequence of energies. It would be interesting to find, if any exist, families of 'static surfaces' (other than spheres and hemispheres) satisfying and study the variance asymptotics for these. In the setting of higher-dimensional standard flat tori T d for d ≥ 4, to the best of our knowledge the only result available in the literature is the expectation (1.8). Higher dimensions and intersections against higherdimensional toral submanifolds are likely to be interesting problems at the interface of number theory and geometry.

Acknowledgements
The author worked on this project mainly during his PhD studies, under the supervision of Igor Wigman. The author is very grateful to Igor for suggesting this very interesting problem, and for his guidance, insightful remarks and corrections. Many thanks to Maurizia Rossi for helpful discussions. Many thanks to an anonymous referee for helpful corrections on a previous version of this manuscript. The author was supported by a Graduate Teaching Scholarship, Department of Mathematics, King's College London.

Background
Consider a random field P defined on a parameter set T ⊂ open R n and taking values in R. 5 We always assume that the P (t), called the realisations or sample paths of our random field, are almost surely continuous in t. A random field P = (P t ) t , t ∈ T , is Gaussian if, for all k = 1, 2 . . . and every t 1 , . . . , t k ∈ T , the random vectors (P (t 1 ), . . . , P (t k )), called finite-dimensional distributions of P , are multivariate Gaussian. A centred (i.e., mean 0) Gaussian field may be completely described by its covariance function (see Kolmogorov's Theorem [13, section 3.3] The arithmetic random wave (1.4) is a centred Gaussian stationary random field, in the sense that its covariance function is a centred Gaussian random field, with unit variance and covariance function As mentioned in section 1.5, for a process p (i.e., a random field with a one-dimensional parameter set) satisfying appropriate assumptions, moments of the number of zeros may be computed via Kac-Rice formulas [2,13,1]. More generally, for a random field defined on R n , there exist under certain conditions Kac-Rice formulas computing the moments of the Hausdorff measure h n−1 of its (n − 1-dimensional) zero set [2, Chapter 6] [9, Section 3]. For a real-valued random field P : Σ → R defined on a smooth surface Σ, consider its nodal length, The formulas of [2] are not applicable to this case, since in particular Σ is not a set of full measure in R 3 (also recall Example 1.6). Given a random field X : R 3 → R and a surface Σ ⊂ R 3 satisfying appropriate assumptions, we derive Kac-Rice formulas for the first and second moments of the nodal length of P = X | Σ (also see [28,Theorem 5.3

Co-area formula
Firstly, we require a general (i.e. concerning manifolds) version of the coarea formula.
and ϕ : X → Y be a Lipschitz map. We denote Jϕ the Jacobian 6 of ϕ. Then Next we require the definition of surface gradient. For a differentiable map ψ : R 3 → R and a point x ∈ R 3 , we employ the standard notation for the gradient of ψ in R 3 . 6 The square root of the sum of the squares of the k × k minors of the k × n matrix Definition 2.2. Fix a surface Σ ⊂ R 3 as in Definition 1.1. For every point σ ∈ Σ, denote T σ (Σ) the tangent plane to the surface at σ. Given a differentiable map G : R 3 → R, consider its restriction to Σ. At each point σ ∈ Σ we define the surface gradient The surface gradient gives the direction of maximal variation of G at σ (for further details, see e.g. [1, chapter 7] or [15, section 2.5]). We record that JG(σ) = |∇ Σ G(σ)|.
where χ is the indicator function of the interval [−1, 1].
We defer the proof of Proposition 2.3 to Appendix A.

Kac-Rice for the expectation
Proposition 2.4 (Kac-Rice for the expected length). Let X : R 3 → R be a Gaussian random field having C 1 paths, and Σ ⊂ R 3 a surface as in Definition 1.1. Define L(X , Σ) and L ǫ (X , Σ) as in (2.2) and (2.4) respectively. Suppose that, for all σ ∈ Σ, the distribution of the random variable X (σ) is non-degenerate. Moreover, assume that L ǫ is uniformly bounded, and that, for every σ ∈ Σ, the quantity is bounded as a function of σ uniformly in ǫ. Then we have where K 1;Σ : Σ → R, and φ X (σ) is the probability density function of the random variable X (σ).
Proof. We follow [ We take expectations on both sides of (2.3): As L ǫ is uniformly bounded by assumption, we may apply the dominated convergence theorem: By Fubini's Theorem, By assumption, the quantity (2.5) is bounded as a function of σ uniformly in ǫ, hence we may apply the dominated convergence theorem to exchange the order of the limit and the integral over Σ: Via dominated convergence as ǫ → 0, hence (2.6).
Let us compare the quantity K 1;Σ (2.7) to the zero density function (or first intensity) K 1 : R 3 → R, of the random field X : R 3 → R: the zero density function has the gradient ∇ of R 3 , in place of the surface gradient ∇ Σ , in its definition. We will call K 1;Σ the "zero density of X | Σ ", as a generalisation of K 1 to random fields defined on a manifold.

The proof of Proposition 1.2
Recall the expression of the arithmetic random wave F (1.4). The following lemma, that will be proven in appendix A, shows that F satisfies one of the hypotheses of Proposition 2.4.
Lemma 2.5. Let F = F m be an arithmetic random wave, and Σ ⊂ T 3 a surface as in Definition 1.1. Then we have We now compute the zero density K 1;Σ for arithmetic random waves.
Lemma 2.6. Given the random field X = F , define the function K 1;Σ as in (2.7). Then we have Before proving Lemma 2.6, we will complete the proof of Proposition 1.2.
Proof of Proposition 1.2. We need to show that the hypotheses of Proposition 2.4 hold for the random field X = F . First, the non-degeneracy condition is met, as F is unit variance. The boundedness of L ǫ (F, Σ) was shown in Lemma 2.5.
Since the quantity is bounded as a function of σ independent of ǫ [35, proof of Proposition 4.1] and since, clearly, |∇ Σ F (σ)| ≤ |∇F (σ)|, we also obtain the boundedness of in σ independent of ǫ. Substituting the expression (2.8) of the zero density K 1;Σ into (2.6) yields Proof of Lemma 2.6. We write the zero density function (2.7) for the Gaussian field F : Define the vector field Since F (σ) and a(σ) are independent (as F has unit variance), we may rewrite (2.9) as One has ∇F (x) ∼ N (0, M I 3 ) for each x ∈ T 3 [35, (4.1)]. Since at each σ the surface gradient a is the projection of ∇F onto T σ Σ (see Definition 2.2), one has so that universally

Kac-Rice for the second moment
Proposition 2.7 (Kac-Rice for the second moment). Let X : R 3 → R be a Gaussian random field having C 1 paths, and Σ ⊂ R 3 a surface as in Definition 1.1. Define L(X , Σ) and L ǫ (X , Σ) as in (2.2) and (2.4) respectively.
Assume that L ǫ is uniformly bounded, and that for almost all σ, σ ′ ∈ Σ, the quantity Suppose further that for almost all σ, σ ′ ∈ Σ, the distribution of the random vector and φ X (σ),X (σ ′ ) is the joint probability density function of the random vector (X (σ), X (σ ′ )).
Proof. We follow [35,Proposition 5.2]. By Ylvisaker's theorem [2, Theorem 1.21], one has P(∃σ ∈ Σ : X (σ) = 0, ∇ Σ X (σ) = 0) = 0. Therefore (recall (2.4)), As L ǫ is uniformly bounded by assumption, we may apply the dominated convergence theorem: By Fubini's theorem, Next, we exchange the order of taking the limit and the integration over Σ 2 , using the boundedness of (2.11) uniformly in ǫ 1 , ǫ 2 together with the dominated convergence theorem: an upper bound for almost all σ, σ ′ is sufficient, as changing the values of a function on a set of measure zero has no impact on integrability or value of the integral of the function. Finally, as we obtain The quantityK 2;Σ (2.12) will be called the "two-point correlation function of X | Σ ", as a generalisation of the two-point function or second intensitỹ to random fields defined on a manifold. Proposition 2.7 is a Kac-Rice formula for the usual second moment, and not for the factorial second moment. This is consistent with Kac-Rice formulas for random fields defined on R n [2, p. 134], as may be seen for instance by comparing [2, Theorem 6.3] with [2, Theorem 6.9].
3 Approximate Kac-Rice formula: proof of Proposition 1.7 In the present section we establish the approximate Kac-Rice formula of Proposition 1.7. We will require a few technical lemmas: the proof of these is deferred to appendix A.
3.1 An expression for the (scaled) two-point correlation function K 2;Σ Recall our notation (2.10) for the surface gradient and let a ′ = a(σ ′ ). One has where − → n = (n 1 , n 2 , n 3 ) is the unit normal to the surface at the point σ. At least one coordinate of − → n , w.l.o.g. n 3 , is non-zero hence We may thus writẽ In order to rewrite (3.2) in a more convenient way, we will need some extra notation. Bearing in mind We will denote Ω ′ = Ω(σ ′ ). Recall the expression (2.1) for the covariance function r.
where for j = 1, 2, 3 we have computed the partial derivatives be the (symmetric) Hessian matrix of r. We also define the matrix Note that and that H(σ ′ , σ) = H(σ, σ ′ ).
Lemma 3.2. The covariance matrix Φ of the Gaussian random vector is given by where M = 4π 2 m/3, Ω is given by (3.3), and D, H, L are as in Definition 3.1.
We may now rewriteΘ We have established the following result.

Asymptotics for K 2;Σ
We will need the following lemma, to be proven in appendix A.
Lemma 3.5. The entries of X, X ′ , Y, Y ′ are uniformly bounded (with respect to σ, σ ′ ): To write an asymptotic expression for the scaled two-point function, we need the Taylor expansion of a perturbed 4 × 4 standard Gaussian matrix to the second order. The case where X ′ = X and Y ′ = Y , to the fourth order, was treated in [26, Lemma 5.1] (4 × 4 matrix) and [3, Lemma 5.8] (6 × 6). In [26,3] the expansion up to order four is needed in light of a cancellation phenomenon known as 'arithmetic Berry cancellation'; in this work we need only an expansion to order two.
Lemma 3.6. Supposê is positive definite with real entries, and the 2 × 2 blocks X, X ′ , Y, Y ′ are symmetric. Then The proof of Lemma 3.6 will be given in appendix A. Assuming it, we obtain the following asymptotic for the scaled two-point function K 2;Σ .
Proposition 3.7. For σ, σ ′ such that r(σ, σ ′ ) is bounded away from ±1, we have the following asymptotic for the (scaled) two point correlation function: with X, X ′ , Y, Y ′ as in Definition 3.3 7 .
Proof. By assumption, r(σ, σ ′ ) is bounded away from ±1, hence 1 √ 1−r 2 = 1 + 1 2 r 2 + O(r 4 ). We apply Lemma 3.6 to expand the precise expression of the two-point function given by Lemma 3.4: hence the result of the present proposition. 7 Here and elsewhere we will use the shorthand O(A) for O(tr(A)).

Statement of further auxiliary lemmas
The present section is dedicated to stating lemmas needed to prove Proposition 1.7. The proofs of the lemmas will follow in section 6.
Lemma 3.8. For k ≥ 0, we define the k-th moment of the covariance function r (2.1) on the surface Σ, Assume Σ is of nowhere zero Gauss-Kronecker curvature. Then for every ǫ > 0 we have the upper bound (3.10) The proof of Lemma 3.8 will be given in section 6.
Lemma 3.9. We have the following upper bounds: The proof of Lemma 3.9 will be given in section 6.

The proof of Proposition 1.7
We claim that the hypotheses of Kac-Rice (Proposition 2.7) hold. The non-degeneracy condition is met since for almost all σ, σ ′ ∈ Σ. Moreover, by Lemma 2.5, one has the uniform bound Thanks to [35,Lemma 5.3] and the fact that |∇ Σ F (σ)| ≤ |∇F (σ)|, one has for almost all σ, σ ′ ∈ Σ, where the implied constant is independent of ǫ 1 , ǫ 2 . Therefore, one may exchange the order of taking the limit and the integration over Σ 2 in Proposition 2.7. The hypotheses of Kac-Rice are thus all verified, hence We will show that Proposition 3.7 applies 'almost everywhere' in the sense that r is small outside of a small set. Since the surface Σ is compact and regular, one may write where γ q : U q ⊂ R 2 → Σ, 1 ≤ q ≤ Q, are finitely many parametrisations as in (1.16), and the union is disjoint save for boundary overlaps. These overlaps are a finite union of smooth curves possibly together with a set of points, therefore the intersection with For each q, consider the smallest rectangle U q ⊇ U q with sides parallel to the coordinate axes of R 2 . Partition (with boundary overlaps) U q into small squares U q,p of side length δ = c 0 / √ m for some small c 0 > 0. This means U q is the disjoint union (with boundary overlaps) of the U q ∩ U q,p =: U q,p . Each γ q is bijective, thus each Σ q is the disjoint union of the Σ q,p := γ q (U q,p ).
To simplify the notation, we re-label the indices q, p to a single index i and write Σ = ∪ i Σ i . The set Σ × Σ is thus partitioned (with boundary overlaps) into regions Σ i × Σ j =: V i,j . Definition 3.10. We say the region V i,j is singular if there are points σ ∈ Σ i and σ ′ ∈ Σ j s.t. |r(σ, σ ′ )| > 1/2. The union of all singular regions is the singular set S. where R k (m) is the k-th moment (3.9) of the covariance function r on the surface Σ.
Proof. As r/ √ m is a Lipschitz function, with constant independent of m, everywhere on V i,j , provided c 0 is chosen sufficiently small. An application of the Chebychev-Markov inequality now yields the statement of the present lemma.
Outside of S Proposition 3.7 applies so that we may rewrite (3.11) as To control the third summand in (3.12) we state the following auxiliary result.
Lemma 3.12. We have the bound The proof of Lemma 3.12 will follow in appendix A. By Lemmas 3.8, 3.9, and 3.12 we may rewrite (3.12) as Changing the domain of integration to Σ 2 carries an error of m · R 4 (m) thanks to Lemmas 3.5 and 3.11. Subtracting the expectation squared (Proposition 1.2) completes the proof of the approximate Kac-Rice formula Proposition 1.7.

Lattice points on spheres 4.1 Background
To estimate the second and fourth moments of the covariance function r and of its derivatives (in sections 5 and 6 respectively), we will need several considerations on lattice points on spheres √ mS 2 . An integer m is representable as a sum of three squares if and only if it is not of the form 4 l (8k + 7), for k, l non-negative integers [23,14]. The total number of lattice points N m = r 3 (m) oscillates: it is unbounded but vanishes for arbitrarily large m. We have the upper bound [7, section 1] The condition m ≡ 0, 4, 7 (mod 8) is equivalent to the existence of primitive lattice points (µ (1) , µ (2) , µ (3) ), meaning µ (1) , µ (2) , µ (3) This lower bound is ineffective: the behaviour of r 3 (m) is not completely understood [7, section 1]. Given a sphere C ⊂ R 3 and a point P ∈ C, we define the spherical cap T centred at P to be the intersection of C with the ball B s (P ) of radius s centred at P . We will call s the radius of the cap. We shall denote the maximal number of lattice points contained in a spherical cap T ⊂ √ mS 2 of radius s.  Linnik conjectured (and proved under GRH) that the projected lattice points E m become equidistributed as m → ∞, m ≡ 0, 4, 7 (mod 8). This result was proven unconditionally by Duke [16,17] and by Golubeva-Fomenko [22] following a breakthrough by Iwaniec [24]. As a consequence, one may approximate a summation over the lattice point set by an integral over the unit sphere.
Define the probability measures on the unit sphere, where δ x is the Dirac delta function at x. By the equidistribution of lattice points on spheres, the τ m converge weak-* 8 to the uniform measure on the unit sphere: as m → ∞, m ≡ 0, 4, 7 (mod 8).
Definition 4.4. For s > 0, the Riesz s-energy of n (distinct) points P 1 , . . . , P n on S 2 is defined as Bourgain, Sarnak and Rudnick computed the following precise asymptotics for the Riesz s-energy of the projected lattice points E m ⊂ S 2 (4.2).
Take j = 3 (by the symmetry, (4.5) will be independent of the choice of j). The integral in (4.5) may be rewritten as 1 4π Similarly, for k = 2, Lemma 4.3 yields The latter integral may be computed via the same method as the case k = 0 and equals 1/15.
We require an estimate for the second factor on the RHS of (4.11). Consider the geometric picture: the vector v lies on the sphere centred at µ 1 + µ 2 + µ 3 of radius √ m, and also in the difference set of the two balls centred at the origin of radii B, A. Therefore, µ 4 lies on a spherical cap of radius 2B of a sphere of radius √ m, hence the bound via Lemma 4.1. Replacing (4.12) into (4.11), Third range: |v| ≥ B. Here we have (4.14) Collecting the estimates (4.10), (4.13) and (4.14), we obtain Bearing in mind (1.2), the optimal choice for the parameters is (A, B) = (N 5/8 , N 11/16 ), so that one has the estimate as claimed.
5 Second moment of r and of its derivatives: proof of Theorem 1.3 The present section is dedicated to proving Theorem 1.3.

Statement of estimates
We recall that A is the area of the surface Σ, r the covariance function (2.1), and Definition 3.3 for X, X ′ , Y, Y ′ .
Lemma 5.1. Assume Σ is of nowhere zero Gauss-Kronecker curvature. Then we have the following estimates: Lemma 5.1 will be proven in section 5.2. We define Assume Σ is of nowhere zero Gauss-Kronecker curvature. Then we have the following estimate: Proof. Firstly, using Lemma 4.6, we estimate the summation By substituting (5.6) into (5.4), we obtain We may now establish the asymptotics for the nodal intersection length variance.
The estimate for H given by Lemma 5.3 now implies (1.14).
In the rest of section 5 we prove Lemmas 5.1 and 5.2. Firstly, we will require a bound for oscillatory integrals on a surface.

Proof of Lemma 5.1
We square the covariance function (2.1) and integrate it over Σ 2 : the diagonal terms equal We bound the off-diagonal terms by applying Proposition 5.4 followed by Proposition 4.5 with s = 2 − ǫ: This completes the proof of (5.1).
Next, we show (5.2), (5.3) being similar. The proof of the following preliminary lemma is deferred to Section 6.
(5.10) By Definition 3.3, where the matrices D, L are as in Definition 3.1, Ω as in (3.3), and Q as in (3.6). We separate the domain of integration into the singular set S of Definition 3.10 and its complement: where we used the uniform boundedness of X given by Lemma 3.5. On Σ 2 \ S the covariance function r is bounded away from ±1, so that We have where in the last equality we noted that LQ 2 L T Ω = I 3 (5.14) by (3.5). Inserting (5.13) into (5.12) and integrating over Σ 2 \ S, By Definition 3.1, The uniform bound implies DΩD T ≪ M . We may then rewrite the main term of (5.15) as We substitute (5.18) into (5.15), and then (5.15) into (5.11) to obtain Inserting the estimates (5.7) and (5.8) into (5.19), and bearing in mind Lemmas 3.11 and 3.8 concludes the proof of (5.2), (5.3) being similar.
The summation over the 4-correlations is where we applied (4.8). By Proposition 5.4, the remaining summation is bounded by By Lemma 4.7, the latter summation has the upper bound N 2+5/8+ǫ . It follows that where we applied (1.2).
Proof of Lemma 5.5. One clearly has µ |µ| 2 = mN , hence µ (µ (i) ) 2 = mN 3 for i = 1, 2, 3. (6.1) By the symmetry of the lattice points, one also has µ µ (i) µ (j) = 0 for any i = j. Bearing this in mind, we directly compute the diagonal terms µ = µ ′ of (5.16) to equal giving a contribution of to (5.7). To control the off-diagonal terms µ = µ ′ of DΩD T , we start by applying Proposition 5.4 to the integrals obtaining the bound where we also applied Proposition 4.5. Consolidating the estimates (6.2) and (6.3) completes the proof of (5.7).
We now show (5.8). To treat the diagonal terms in r 2 DΩD T , we use (4.8) and the triangle inequality to obtain (recall the abbreviation µ j := µ 1 + µ 2 + µ 3 + µ 4 ). Next, we consider the off-diagonal terms of r 2 DΩD T . Invoking Proposition 5.4 again, having used Lemma 4.7 in the last inequality. Consolidating the estimates (6.4) and (6.5) completes the proof of (5.8).
To prove (5.9), we start with Definition 3.1, One directly calculates the diagonal terms of the latter expression to equal having used (6.1) in the last step. We control the contribution of the offdiagonal terms of tr (HΩHΩ ′ ) via Propositions 5.4 and 4.5: Substituting (6.6) and (6.7) into (5.22) and recalling the definition of H (5.4) yields (5.9). The computation for (5.10) is very similar to the preceding ones and we omit it here.
Proof of Lemma 3.9. We will prove in detail a few of the bounds, the remaining being similar. By squaring X (Definition 3.3) we obtain Therefore, outside the singular set S (Definition 3.10), where we used (5.14). We may thus write The diagonal terms of the integral on the RHS in (6.9) give a contribution of via (4.8). The off-diagonal terms are bounded via Proposition 5.4 and Lemma 4.7: Inserting the bounds (6.10) and (6.11) into (6.9) (and bearing in mind Lemmas 3.11 and 3.8) yields the desired estimate for tr(X 2 ). The one for tr(X ′2 ) is proven in a similar way.
Let us now prove the bound for tr(Y 4 ), the one for tr(Y ′4 ) being similar. By Definition 3.3, via (5.14). Therefore, Finally, we show the bound for r 2 tr(X), the ones for r 2 tr(X ′ ) and r 2 tr(Y ′ Y ) being similar. As in the previous computations, one writes The required result now follows from (5.8) and Lemmas 3.11 and 3.8.
7 Study of the variance leading constant: proof of Proposition 1.4 Recall A is the area of the surface Σ, − → n (σ) the unit normal at the point σ ∈ Σ, and I the integral (1.13) The goal of this section is to prove Proposition 1.4, i.e., that the sharp bounds with equality if and only if Σ is contained in a plane.
Proof. The integral I is maximised when for every σ, σ ′ ∈ Σ, i.e., when the normal vectors to the surface are all parallel.
We now turn to establishing the lower bound I ≥ A 2 /3. For any probability measure τ on S 2 invariant by reflection w.r.t. the coordinate planes, define the number Lemma 7.2. For any measure τ on S 2 invariant by reflection w.r.t. the coordinate planes, we have Lemma 7.2 will be proven in a moment; assuming it, we may complete the proof of Proposition 1.4.
The upper bound I ≤ A 2 in (1.15) has already been proven in Lemma 7.1.
Let us compare the conditions to obtain the vanishing of the variance leading term in the two-and three-dimensional settings. In the former, this occurs for certain subsets of circles [36,Proposition 7.3] and more generally, for families of static curves [34, appendix F]. We were able to find specific examples of surfaces s.t. the variance leading term vanishes (e.g. spheres and hemispheres): it would be interesting to find, if it exists, a more general family of surfaces satisfying this condition.

A Proofs of auxiliary results
In this appendix, we prove several auxiliary propositions and lemmas. By assumption, G satisfies G(σ) = 0 ⇒ ∇ Σ G(σ) = 0 for every σ ∈ Σ, hence the function y → h 1 (G −1 {y}) is continuous at y = 0, so that, by the fundamental theorem of calculus, lim ǫ→0 L ǫ = lim As the surface gradient ∇ Σ F is the projection of ∇F on T σ Σ, (A.2) In [35,Lemma 3.2], it was shown that the quantity satisfies the uniform bound Substituting the latter bound into (A.2), we obtain the claim of the present lemma.
Next, we compute the Taylor expansion of a perturbed Gaussian covariance matrix, establishing Lemma 3.6. We employ Berry's elegant method [4], rather than computing the various derivatives by brute force, which would result in a longer computation.  We conclude the appendix with the following proof.
Proof of Lemma 3.12. We begin by invoking Lemma 3.4 to writẽ andΘ is given by (3.7). As σ ′ → σ the expectation in (A.6) converges to a positive constant, so that on each diagonal region V i,ĩ via a Taylor expansion. Therefore, By Kac-Rice and the Cauchy-Schwartz inequality, where we also applied Lemma 3.11. The proof of Lemma 3.12 is complete.