The defect of toral Laplace eigenfunctions and Arithmetic Random Waves

We study the defect (or"signed area") distribution of toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, in either random Gaussian scenario ("Arithmetic Random Waves"), or deterministic eigenfunctions averaged w.r.t. the spatial variable. In either scenario we exploit the associated symmetry of the eigenfunctions to show that the expectation (Gaussian or spatial) vanishes. Our principal results concern the high energy limit behaviour of the defect variance.


INTRODUCTION
1.1. Toral Laplace eigenfunctions and Arithmetic Random Waves. Toral Laplace eigenfunctions are an important model in Quantum Chaos that represent the Laplace eigenfunctions on generic manifolds. From the point of view of an investigator interested in the study of their properties, the toral eigenfunctions enjoy two significant privileges over the general case, making them attractive to address, in addition to their own sake, being Fourier sums with particular frequencies. First, its number theoretic ingredient makes them susceptible to methods borrowed from Analytic Number Theory. Second, their (slowly in 2 dimensions) growing spectral degeneracies allow for the study of the "typical" case, whether that means endowing the linear space of Laplace eigenfunctions with the same eigenvalue with a Gaussian probability measure (thus giving rise to "Arithmetic Random Waves"), or otherwise.
Conversely, every real-valued function satisfying the equation (1.3) is necessarily of the form (1.1) for some n ∈ S, and {a λ } λ 2 =n as above.
We denote E n = {(λ 1 , λ 2 ) ∈ Z 2 : λ 2 1 + λ 2 2 = n} to be the representations of n as sum of two squares, or, what is equivalent, E n are all standard lattice points lying on the radius-√ n circle. One may endow this space with a probability measure by assuming that the {a λ } λ∈En are standard (complex) Gaussian 1 i.i.d. save to (1.2), turning {f n } n∈S into a Gaussian ensemble of random fields [22,25], all defined on T 2 , usually referred to as "Arithmetic Random Waves" [16]. Alternatively, f n are unit variance stationary random fields on T 2 , uniquely defined via their covariance function (1.5) r n (x) = r n (y, x + y) := E[f n (y) · f n (x + y)] = 1 N n λ∈En cos(2π λ, x ).

1.2.
Defect. The (total) defect of a smooth, not identically vanishing, function g : T 2 → R, (called "signed area" within the physics literature) is D(g) := Area(g −1 (0, +∞)) − Area(g −1 (−∞, 0)) = The defect of Laplace eigenfunctions was first addressed in the physics literature [4] for random planar monochromatic waves. A precise asymptotic expression for the defect variance, and a Central Limit Theorem was established, along with generic nonlinear functionals, for the ensemble {T l } l≥1 of random Gaussian spherical harmonics [19,20] with mathematical rigour. The T l : S 2 → R is the important ensemble of spherical random fields defined by the covariance functions E[T l (x) · T l (y)] = P l (cos(d(x, y))), where P l (·) are the Legendre polynomials and d(·, ·) is the spherical distance; T l (·) scales asymptotically like Berry's Random Waves around every point of S 2 , the main findings of [19,20] being consistent with [4], up to the said scaling.
We are interested in the defect of f n (·) as in (1.1). We claim that for every such function f n , the corresponding defect (1.7) D(f n ) ≡ 0 vanishes, so the study of D(f n ) trivialises, and, accordingly, below we will pass to subdomains of T 2 . First, if n is odd, then for every λ = (λ 1 , λ 2 ) ∈ E n , necessarily precisely one of λ 1 and λ 2 is odd. Hence, f n changes its sign under the involution τ : T 2 → T 2 mapping · → · + (1/2, 1/2), i.e.
f n (τ x) = −f n (x), which readily implies D(f n ) = 0. Otherwise, if n is even, we may assume w.l.o.g. that 2 n ≡ 2(4), whence for all λ ∈ E n , both λ 1 , λ 2 are odd, and then f n changes its sign under the involution ρ : T 2 → T 2 mapping · → · + (1/2, 0) (or · → · + (0, 1/2)), also yielding D(f n ) = 0. It is therefore essential to pass to, possibly shrinking, subdomains of T 2 , most canonically, the radius-s discs B x (s) ⊆ T 2 centred at x ∈ T 2 , 0 < s < 1/2, and B(s) := B 0 (s), with s = s(n) allowed to depend on n, (possibly s = s(n) → 0). Since Quantum Chaos should exhibit itself above Planck scale s ≫ 1 √ n [2], it makes sense to take, as an example, s = n −1/2+ǫ , or, perhaps, replace the ǫ-power of n with a slower growing function of n (such as a power of log n). Our principal results concern the defect distribution corresponding to both the Arithmetic Random Waves (random Gaussian toral eigenfunctions) in §1. 3 where the normalisation makes D n;s invariant w.r.t. homotheties, and, by the stationarity of f n , the law of D n;s is independent of the centre of the disc (which is why we are may assume that the disc on the r.h.s. of (1.8) is centred). Since, for a given y ∈ T 2 , the law of f n (y) is symmetric around the origin, and H(·) is odd, we have E[H(f n (y))] ≡ 0, and, by inverting the integral on the r.h.s. of (1.8), it is evident that for every n ∈ S and s > 0, Our first principal result asserts that Var(D n;s ) → 0 as long as the ball radius is above the Planck scale, i.e., s · √ n → ∞. Var (D n;s ) · N δ n < +∞.
If one is willing to excise a thin sequence of energies, that is, a subsequence S ′ of S whose relative asymptotic density 3 in S is 0, so that whatever generic energy levels are remaining satisfy certain arithmetic conditions explicated in Theorem 2.5 of §2.2 below, then the asserted rate of decay is significantly more rapid, namely, faster than polynomial in N n . Theorem 1.2. For every ǫ > 0 there exists a subsequence S ′ = S ′ (ǫ) ⊆ S of energy levels of relative density 1, so that, along n ∈ S ′ , the inequality holds for every A > 0. 2 Otherwise both the entries λ 1 , λ 2 are even, which yields that f n is invariant under the involutions · → · + (1/2, 0) and · → · + (0, 1/2), and we may pass from n to n/4.
To the other end, we claim the following lower bound for Var(D n;s ) above Planck scale, valid for all n ∈ S. Theorem 1.3. Let s = s(n) be a sequence of radii so that T := s · √ n → ∞.
a. For every δ > 0 there exists a sufficiently large number A = A(δ) so that If, in addition, 2πT is bounded away from the zeros of the Bessel J 1 function, then (1.12) Var (D n;s ) ≫ 1 T 3 . For comparison of the generic upper bound (1.10) with the lower bounds (1.11) and (1.12) (restricted to the regime s > n −1/2+ǫ all the said bounds hold) one should bear in mind (1.4), i.e. that every arbitrarily small positive power of n dominates every power of N n . It is well known that at infinity, the zeros of the Bessel J 1 function are asymptotic to the arithmetic sequence The a fortiori meaning of the condition postulated by Theorem 1.3b is that 2πT is bounded away by at least ǫ 0 > 0 from the said sequence (1.13), whence the conclusions apply (with constants depending on ǫ 0 ).
1.4. Statement of principal results: spatial defect distribution. Rather than working with a Gaussian random field, we can take a sequence of deterministic eigenfunctions f n of the form (1.1), and study the defect distribution of f n restricted to B x (s), where x is random uniform on T 2 , and s is above Planck scale. That is, given a function f n of the form (1.1), x ∈ T 2 and s > 0, we consider (1.14) Y fn,s (x) := 1 πs 2

Bx(s)
H(f n (y))dy, the defect of f n restricted to B x (s). Such an approach was recently taken by Sarnak [26] and Humphries [13] for modular forms, and Granville-Wigman [12] and Wigman-Yesha [34] for toral Laplace eigenfunctions (1.1), in studying the mass distribution of the respective models, showing, in particular, that if there exist discs observing unproportionately large or small L 2 -mass of f n , then these are not "typical". Of our principal interest here is the distribution of the values of Y fn,s (·) in (1.14) as x distributes randomly uniformly on T 2 ; we denote accordingly the "spatial defect expectation" and the "spatial defect variance" The degeneracy argument identical to the argument we used to establish (1.7) that the total defect of every function (1.1) vanishes, yields that, in general, the spatial defect expectation vanishes precisely, i.e., that In what follows, we will restrict ourselves to Bourgain's class [7] of eigenfunctions B n = f n = λ∈En a λ · e( x, λ ) : ∀λ ∈ E n , |a λ | = 1 and a −λ = a λ .
Our principal result concerning the spatial defect distribution asserts that for generic n ∈ S, and f n ∈ B n a Bourgain class function, the spatial defect variance vanishes uniformly for s slightly above Planck scale. Since Y fn,s is bounded, this is equivalent to the statement that, in the said scenario, the proportion of positive values of f n in "most" discs of radius above Planck scale is asymptotic to 1/2 (see Lemma 4.8 below). Despite that, what seems likely, the proof of the principal result immediately below holds for a more general family of flat eigenfunctions of the type considered in [34] (an event of almost full Gaussian probability), we abandon the possible generality for the sake of the elegance of presentation. That some flatness condition is essential for the defect variance vanishing is asserted in Theorem 1.5 to follow immediately after the announced principal result.
Theorem 1.4. There exists a sequence S ′′ ⊆ S of relative density 1, so that for all ǫ > 0 there exists R = R(ǫ) > 0 and n 0 = n 0 (ǫ) sufficiently large, so that for all n > n 0 with n ∈ S ′′ , holds uniformly for all f n ∈ B n , s > R/ √ n. Equivalently 4 , The arithmetic conditions on a sequence S ′′ as postulated in Theorem 1.4 will be explicated in §2.3 below, as part of Theorem 2.6; they are more restrictive as compared to the subsequence S ′ postulated in Theorem 1.2. Finally, the result on the flatness being of essence for the spatial defect variance vanishing announced above is stated, with radii vanishing arbitrarily slowly (or even fixed small radii). Theorem 1.5. There exists a (thin) sequence S ′′′ ⊆ S, a deterministic sequence {f n } n∈S ′′′ of eigenfunctions (1.1), and numbers γ, ǫ 0 > 0, so that the inequality lim inf n∈S ′′′ Var T 2 (Y fn,Ψ(n) ) > ǫ 0 holds for every function Ψ : Z >0 → (0, min(γ, 1/2)), subject to Ψ(n)n 1/2 → ∞.
Acknowledgements. We are indebted to Zeév Rudnick for many stimulating discussions, and his comments on an earlier version of this manuscript, in particular, pertaining to Lemma 3.3 on Diophantine approximations. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013), ERC grant agreement n o 335141 (I.W. and N.Y.). P.K. was partially supported by the Swedish Research Council (2016-03701).

OUTLINE OF THE PAPER
2.1. Number Theoretic preliminaries. Before we will be able to explain the essence of our arguments we will be required to bring forward some arithmetic aspects of the lattice points E n .
2.1.1. Angular equidistribution of lattice points. First, we are interested in the angular distribution of E n . To this end we define the sequence ν n := 1 N n λ∈En δ λ/ √ n of probability measures on S 1 ⊆ R 2 , indexed by n ∈ S. It is well-known [14,10,11] that generically the angles of E n are equidistributed, i.e. along a sequence {n} ⊆ S of relative density 1, Formally, unrolling the definition of the double limit below yields a slightly different, though equivalent to the above, statement, since it is strongest for R small. where, as usual, " ⇒ " stands for weak- * convergence of probability measures, and dθ 2π is the normalised arc-length measure on the unit circle. However, even under the (generic) assumption N n → ∞, there exist sequences {n} ⊆ S so that ν n ⇒ τ with τ different than dθ 2π ; by definition, τ can be any "attainable" probability measure on S 1 , e.g. the Cilleruelo measure [9] or "intermediate" measures (e.g. measures supported on Cantor set, cf. [16]); for a partial classification see [17,28].
Definition 2.1. For a sequence {n} ⊆ S we say that E n are asymptotically equidistributed if (2.1) holds.

Spectral correlations and quasi-correlations.
One of the key ingredients in [16] was controlling the size of length-6 "spectral correlations set". Given l ≥ 3, the length-l spectral correlation set of the torus is the set of l-tuples of lattice points in E n summing up to 0. Since, unless n is divisible by 4 (whence we can pass to n/4 in place of n), for λ ∈ E n , the number of odd coordinates among λ 1 , λ 2 is 1 or 2 depending on the parity of n (but independent of λ ∈ E n ), for l odd, the correlation sets Since for l = 2k, all the "diagonal" tuples (λ 1 , −λ 1 , . . . , λ k , −λ k ) and their permutations are in P n (l), it implies the inequality #P n (l) ≫ N k n . Conversely, Bombieri-Bourgain [6] proved, among other things, that, given l = 2k even, the inequality Definition 2.2 (Correlation-tame sequences of energies). We say sequence S ′ ⊆ S is correlationtame, if for every l = 2k ≥ 6 even, the inequality (2.4) holds true.
In fact, Bombieri-Bourgain [6] proved a stronger property satisfied by the correlations of E n , with n generic, i.e. that a generic sequence in S satisfies the following axiom F (γ) for some 0 < γ < 1/2. (1) For l ≥ 4, n ∈ S, we say that (λ 1 , . . . , λ l ) ∈ E l n is a minimal correlation, if l j=1 λ j = 0 and no proper subsum of l j=1 λ j vanishes.
(2) For 0 < γ < 1/2 we say that a sequence {n} ⊆ S satisfies the axiom F (γ), if for every l ≥ 4, the number of length-l minimal correlations of E n is at most N γ·l n for n sufficiently big.
As we will deal with moments of r n (·) restricted to shrinking balls, we will find that, for our purposes, the relevant notion is that of quasi-correlations [5] (see (2.9) below). Given n ∈ S, ǫ > 0 and l ≥ 2, the length-l quasi-correlation set is 5 C n (l, ǫ) := (λ 1 , . . . , λ l ) ∈ E l n : 0 < l j=1 λ j < n 1/2−ǫ ; note that, by the definition, P n (l) and C n (l, ǫ) are disjoint. It was shown [5, Theorem 1.4] that, given l ≥ 2 and ǫ > 0, the length-l quasi-correlation set is empty C n (l, ǫ) = ∅ along a generic sequence {n} ⊆ S, and, as it is the case of the correlation set, by a diagonal argument, we may choose a density-1 subsequence {n} ⊆ S, so that along that sequence, for every l ≥ 2, holds true for n sufficiently big (depending on l).
Definition 2.4 (Axiom A(ǫ) on sequences of energies). Given ǫ > 0 we say that a sequence S ′ ⊆ S satisfies the axiom 6 A(ǫ), if for every l ≥ 2, the equality C n (l, ǫ) = ∅ holds for n sufficiently big.

2.2.
Outline of the proofs for Arithmetic Random Waves (theorems 1.1-1.3). Here we assume that {f n } n∈S are the (Gaussian) Arithmetic Random Waves. Since it is possible to derive the identity (cf. Lemma 3.1) a straightforward manipulation with the definition (1.8) of D n;s and inverting the order of integration, upon bearing in mind the stationarity of f n , yields the following precise expression for the defect variance: Now we Taylor expand the arcsine around the origin (note that the series converges absolutely at the endpoints t = ±1) where all the (explicit) a k > 0 are positive, and substitute into (2.6) to relate between the defect variance and the moments of the covariance function restricted to B(s): We may in turn exploit the additive structure (1.5) to relate the said odd moments of r n (·) to the spectral correlations (and, implicitly, the quasi-correlations) defined in §2.1.2: with J 1 (·) the Bessel J function of the first order, so that to relate the defect variance to the spectral correlations and quasi-correlations (where, to obtain (2.9), we separate the diagonal and use the observation (2.3)). One may then substitute (2.9) into (2.8) to obtain a more explicit expression for Var(D n;s ), an absolutely convergent infinite series over all (2k + 1)-tuples of lattice points. If we assume further, that s = n −1/2+ǫ (say), and a sequence {n} ⊆ S satisfies the A(δ) axiom with some 5 Mind the slight abuse of notation as compared to [5] 6 Mind again an abuse of notation compared to [5] δ < ǫ, then all the summands on the r.h.s. of (2.9) are formally decaying like a (small) power of n, faster than any power of N n (see (1.4)).
There is a subtlety with this outlined approach though, as controlling the decay rate in this infinite series uniformly seems very difficult (if possible at all). Instead, we will only control finitely many summands and bound the contribution of the higher moments. With this approach, we will encounter the odd moments of the absolute value |r n (·)| of the covariance rather than the moments of the covariance, that we will reduce to a moment of higher order via Cauchy-Schwarz. Theorem 1.1 is the result of such an application when capping the series at the first degree Taylor approximation of the arcsine (2.7), whereas Theorem 1.2 caps it at an arbitrarily high degree Taylor approximation, depending on the required A > 0 in (1.10), while also appealing to the correlation-tame property of a generic sequence of energies. We will be able to prove the following result, which, since the claimed sequence S ′ is generic, thanks to the results mentioned in §2.1.2, clearly implies Theorem 1.2.
Theorem 2.5 (Theorem 1.2 with control over S ′ (ǫ)). Let ǫ > 0 be given, and assume that S ′ ⊆ S is a sequence of energy levels satisfying the axiom A(δ) with some δ < ǫ, and is correlation-tame. Then the conclusions of Theorem 1.2 hold, i.e., along n ∈ S ′ , For the lower bounds in Theorem 1.3 one also starts from (2.8) and (2.9). Indeed, since the Taylor coefficients a k in (2.8) are all positive, and, in hindsight, so are all the moments (2.9) of r n (·), it is sufficient to bound any of these from below. If T := s · √ n happens to be bounded away from zeros of the Bessel J 1 function, this readily yields the bound (1.12) of Theorem 1.3. Most of our argument takes upon the opposite situation when T approaches one of the Bessel J 1 zeros, whence we need to rule out the, a priori unlikely, possibility of all the terms 2πs · 2k+1 j=1 λ j conspiring around the Bessel zeros. To resolve this situation we exploit the higher order Taylor approximates, whence appealing to the deep W. Schmidt's simultaneous Diophantine approximation theorem [31], for example, approximating √ 5 by rational number for k = 1 or √ 13 and √ 17 for k = 2; to attain 1 T 3+δ as in (1.11) we will need to focus on arbitrarily high k. Instead of using such a powerful result as in [31], one can try to significantly soften our techniques by bounding away from integers the values of the linear form L : R K → R given by x j √ p j , with a collection of distinct primes p j ≡ 1(4) of our choice. An application of Khintchine's transference principle [15] (see also [32, Theorem 5C on p. 99-100]) with Liouville's bound Lemma 1A on p. 151], yields information on the simultaneous approximation of { √ p j } by rational numbers. Unfortunately, the exponent, resulting from such an application, grows to infinity with K, which, to our best knowledge, undermines any attempt of the described type, and we thereby abandon it in favour of appealing to [31].

2.3.
Outline of the proofs for spatial fluctuations (Theorem 1.4). By a simple manipulation with the defect definition (1.14) and integration order exchange it is straightforward to derive the expression for the spatial defect variance, where W is a certain weight function ("circle-circle intersection function") supported on [0, 2], and is C 1 on (0, 2). It is conceivable that the asymptotic vanishing of Var T 2 (Y fn,s ) follows by a direct analysis of the r.h.s. of (2.10). However it seems very difficult, as the appearance of H(·) on the r.h.s. of (2.10) does not allow us to capitalise on the special additive structure (1.1) of f n , especially, in light of the discontinuity of H(·) at the origin (so, for example, Taylor expanding H(·) around the origin is problematic).
We abandon such a direct approach, and instead notice that, since the random variable Y fn,s is bounded (by 1), the variance Var T 2 (Y fn,s ) asymptotically vanishing is equivalent to Y fn,s asymptotically vanishing with high probability (i.e. for "most" of the ball centres on the torus), and recall that, under certain flatness conditions on f n (certainly satisfied by all f n ∈ B n ) and arithmetic conditions on n (in the spirit of the ones given in §2.1.2 above), f n (·) exhibits [7,8] Gaussian spatial value distribution when averaged over the whole torus. Using these "de-randomisation" techniques we will be able to prove the result to follow immediately; unlike the results of [7,8] (and [29]), this is a secondorder result (as opposed to a first order one). Moreover, since, unlike [7,8], the Gaussian input for Theorem 2.6 is not inherently contained within its statement, it seems that a more direct approach might be possible for proving Theorem 2.6. Recall axiom F (γ) in Definition 2.3, and lattice points equidistribution in Definition 2.1.
Theorem 2.6 (A variant of Theorem 1.4 with control over S ′′ ). Let S ′′ ⊆ S be a sequence of energy levels satisfying the axiom F (γ) for some γ ∈ (0, 1/2), and assume further that the corresponding E n are asymptotically equidistributed. Then the conclusions of Theorem 1.4 apply along S ′′ , i.e.
Theorem 1.4 is a direct consequence of Theorem 2.6, because axiom F (γ) holds with some γ ∈ (0, 1/2) for "generic" n ∈ S, and E n is asymptotically distributed for "generic" n ∈ S in the sense of Definition 2.3. The proof of Theorem 2.6 proceeds in three steps. First, we reduce proving (2.11) uniformly for s > R/ √ n to proving for s = R/ √ n only, via an analogue of the Geometric-Integral Sandwich, first introduced in [33,21], adapted to our settings. Next, we exploit the said spatial Gaussianity of f n (·) in order to reduce the variance vanishing to the analogous result for the limit random field, which, by the equidistribution assumption for E n of Theorem 2.6, is the Gaussian random field of planar isotropic monochromatic waves (it is "Berry's Random Wave Model", uniquely defined by its covariance function J 0 ( x )).
It then remains to evaluate the variance of the defect for the limit Gaussian random field restricted to a compact domain (e.g. the unit square), which, in spirit, is already contained in [19] (and predicted by [4]), where a rapid decay rate is asserted. This result is the only use of the equidistribution assumption, and it should be not too technically demanding to remove this assumption, as long as some non-degeneracy for the limit Gaussian field is imposed (e.g. it cannot include the most degenerate "Cilleruelo" case), though it benefits us in no way if we are only interested in a density-1 sequences of energy levels. Our main result (2.11) is ineffective in terms of rate of decay for Var T 2 (Y fn,s ), as the convergence of the spatial distribution of f n to the Gaussian is ineffective.

2.4.
Outline of constructing functions with non-vanishing defect variance (Theorem 1.5). The prevailing symmetry obstruction, dictating that for the standard torus, the total defect of any Laplace eigenfunction vanishes precisely does not persist for the non-standard tori. We exploit the hexagonal torus, so that to construct a single Laplace eigenfunction with total defect non-vanishing, and scale it to obtain a sequence of eigenfunctions of arbitrarily high energy, with defect growing on large fragments of the torus, above the Planck scale. We then mimic that situation on the standard torus, by appealing to the Pell equation x 2 − 3y 2 = 1, yielding solutions approximating the hexagonal toral eigenfunctions on the standard torus.
2.5. Outline of the paper. Section 3 is dedicated to giving the proofs for all the results concerning the defect of the Arithmetic Random Waves (theorems 1.1-1.3), appealing among the rest to Diophantine approximations. In section 4 Bourgain's de-randomization method will be invoked to prove Theorem 1.4 dealing with the spatial defect variance vanishing for the flat functions. Finally, a sequence of "esoteric" non-flat functions with spatial defect variance non-vanishing will be constructed in section 5, by first constructing eigenfunctions with the analogous properties defined on the hexagonal torus (as opposed to the standard torus).
3. THE DEFECT OF ARITHMETIC RANDOM WAVES: PROOF OF THEOREMS 1.1-1.3 3.1. Preliminary lemmas. Let f n (·) be the Arithmetic Random Wave corresponding to (1.1), so that f n (·) is a unit variance stationary Gaussian random field with covariance function (1.5). We first establish the precise expression (2.6) for the variance of the defect D n;s . Proof. It is a well-known fact (see, e.g., [23,24]) that every bivariate centred Gaussian random vector (X, Y ) with covariance matrix Hence, the identity (2.5) follows letting X = f n (x), Y = f n (y). By the vanishing of the defect expectation (1.9), we have Changing the order of expectation and integration in (3.1) together with the identity (2.5) gives the desired formula for the defect variance.
As we will see below, the defect variance Var (D n;s ) is intimately related to the (restricted) moments of the covariance function r n (·). The following lemma gives a useful arithmetic formula for these moments.

Upper bounds.
We now turn to prove the upper bounds for Var (D n;s ). We begin with the proof of Theorem 1.1.

Proof of Theorem 1.1. By Lemma 3.1 and the elementary bound arcsin
which, using the bound, for all s > n −1/2+ǫ . We find that the contribution from the first integral on the r.h.s. of (3.3) is ≪ n −3ǫ .
We next evaluate the second integral on the r.h.s. of (3.3). By Lemma 3.2, we have where we used the fact that (λ 1 , λ 2 ) ∈ P n (2) if and only if λ 1 = −λ 2 , and in particular #P n (2) = N n , and the symmetry λ ∈ E n ⇐⇒ −λ ∈ E n . Again using the bound (3.4) we have and therefore, for any 0 < η < 1/2, we have (3.6) We estimate the sums on the r.h.s. of (3.6) separately. The second sum on the r.h.s. of (3.6) can be bounded trivially: whereas the first sum on the r.h.s. of (3.7) is the number of "close-by pairs", bounded in [12] (see Theorem 1.8 there and the remark following it) by Let 0 < δ < 4ǫ, and write δ = τ η where τ < 4 and η < ǫ. Then (3.9), together with (3.3), the bound (1.4), and the previous bound on the first integral on the r.h.s. of (3.3), gives Var (D n;s ) ≪ N −δ n uniformly for all s > n −1/2+ǫ , completing the proof of Theorem 1.1.
We now prove Theorem 2.5 which, as argued above, immediately implies Theorem 1.2.
Substituting the bound (3.13) together with the bound (3.4) into (2.9), we get that uniformly for s > n −1/2+ǫ . We can now use (3.14) to bound the summation in the variance formula (3.12), which gives To control the (2K + 3)'th moment of the absolute value of r n (·), we use the Cauchy-Schwarz inequality to discard the absolute value: Since S ′ is correlation-tame (Definition 2.2), we have #P n (4K + 6) ≪ K N 2K+3 n . This, together with (3.17) and the estimate (3.4), yields By the lower bound (3.13), we have uniformly for s > n −1/2+ǫ . Substituting the bound (3.19) into (3.18) and bearing in mind (1.4) gives Finally, we substitute the bound (3.20) into (3.16), and then into (3.15). Using again (1.4), we get that This completes the proof of Theorem 2.5, since K can be taken arbitrarily large.

Lower bound.
In order to prove the lower bound for Var (D n;s ) stated in Theorem 1.3, we will require a result on Diophantine approximation by multiples of square roots of prime numbers. For t ∈ R, we denote t to be the distance of t to the nearest integer number, and let The proof of Lemma 3.3 will invoke two classical results from the theory of Diophantine approximation: Besicovich's theorem on the linear independence over Q of the square roots of distinct square-free positive integers, and Schmidt's theorem on simultaneous Diophantine approximation, that, for the reader's convenience, we cite next, in the form used subsequently.  Proof of Lemma 3.3. By Theorem 3.4, the elements of the set {1} ∪ √ p : p ∈ P K are linearly independent over the rationals. Since #P k ∼ K 2 log K as K → ∞, the bound (3.22) follows from Theorem 3.5.
We are finally in a position to prove Theorem 1.3.

SPATIAL DEFECT DISTRIBUTION: PROOF OF THEOREM 1.4
Recall that Theorem 1.4 follows at once from its more explicit variant, Theorem 2.6, whose proof is the ultimate goal of this section. 4.1. Proof of Theorem 2.6. The following proposition is seemingly weaker, or less general, compared to Theorem 2.6, as it only allows for radii s = R √ n with R → ∞ growing slowly, instead of a uniform statement for all s > R/ √ n as in (2.11). However, we will be able to infer the more general result, using the elegant Integral-Geometric Sandwich in Proposition 4.2 below, inspired to high extent by its counterpart introduced by Nazarov-Sodin [33,Lemma 1] for the sake of counting the number of nodal components (see also [27,Lemma 3.7] and [21, Lemma 1]). It seems a priori counter-intuitive that it is "easier" to first establish the spatial defect variance vanishing for smaller radii than bigger ones. Our explanation of the said surprise is that the asymptotic Gaussianity w.r.t. the spatial variable holds at Planck scale only (or logarithmically above it [30]), rather than at all scales above it.
Proposition 4.1 (Planck scale spatial defect distribution). Let S ′′ ⊆ S be any sequence of energy levels satisfying the assumptions of Theorem 2.6. Then for every ǫ > 0 there exists R 0 = R 0 (ǫ) > 0 sufficiently large so that for all R > R 0 there exists a number n 0 = n 0 (R, ǫ) sufficiently large so that for all n > n 0 , the inequality holds uniformly for f n ∈ B n . Equivalently, The following proposition asserts the aforementioned Integral Geometric Sandwich; unlike the original inequality, it contains an error term. Recall that the local (normalized) defect of a function an eigenfunction f n as in (1.1) restricted to a radius-s ball around x ∈ T 2 is given by (1.14).

Proposition 4.2 (Integral Geometric Sandwich)
. For every f n of the form (1.1), and 0 < r 1 < r 2 , the asymptotic estimate Y fn,r 1 (y)dy + O r 1 r 2 holds, with constant associated to the 'O ′ -notation absolute.
Proof of Theorem 2.6 assuming propositions 4.1-4.2. Let ǫ > 0 be given. First, we apply Proposition 4.1 to obtain a number R 0 = R 0 (ǫ) so that for all R > R 0 there exists a number n 0 = n 0 (R, ǫ) so that for n > n 0 with n ∈ S ′′ , one has uniformly for all f n ∈ B n . We define and claim that with this choice of R, the conclusion of Theorem 2.6 holds, where the corresponding n 0 = n 0 (R 0 + 1, ǫ), depending on ǫ only, is the one we received as the output from the application above of Proposition 4.1. For this particular choice of the parameters, the inequality (4.3) reads valid for all n ∈ S ′′ , n > n 0 and f n ∈ B n . To validate our claim we are to prove that for all n > n 0 with n ∈ S ′′ , the inequality (4.6) Var T 2 (Y fn,s ) < ǫ holds for all s > R √ n . Now, we invoke the Integral Geometric Sandwich of Proposition 4.2, with r 2 = s > R/ √ n and (4.7) thanks to (4.7). We assume that R 0 is sufficiently large so that the error term on the r.h.s. of (4.8) take the absolute value of both sides of (4.8), and apply the triangle inequality to conclude that We then integrate both sides of (4.9) w.r.t. x ∈ T 2 to yield and invoke (4.5) together with Cauchy-Schwarz inequality, that gives (recalling that the spatial expectation vanishes identically, see (1.15)) (4.10) Finally, the inequality (4.10) certainly implies (4.6), since |Y fn,s (x)| ≤ 1 (again, upon recalling (1.15)), which, as it was mentioned above, is sufficient to infer the statement of Theorem 2.6.

Integral Geometric Sandwich: Proof of Proposition 4.2.
Proof. We start with the integral on the r.h.s. of (4.2), and use the definition (1.14) to write H(f n (z))dzdy, and aim at reversing the order of the integrals on the r.h.s. of (4.11). We have Vol (B z (r 1 ) ∩ B x (r 2 )) dz. H(f n (z)) · V x,z (r 2 , r 1 )dz, and we notice that

4.3.
Auxiliary results towards the proof of Proposition 4.1. We denote Berry's random monochromatic isotropic waves g : R 2 → R defined on a probability space (Ω, Σ, Pr), i.e. for ω ∈ Ω the corresponding sample function g(·) = g ω (·) are distributed as a centred Gaussian random field uniquely determined via Kolmogorov's Theorem by its covariance function where J 0 is the Bessel J function of order 0. Proposition 4.3 immediately below asserts that locally, the functions f n ∈ B n , appropriately scaled, converge to g(·) around a random spatial variable on the torus, understood as random fields. It is the heart of Bourgain's de-randomization method, originally in [7], and is a restatement of what turned out to be the key technical propositions in [8], in the precise form used in that manuscript. To state this result, given a function f n ∈ B n , we introduce the function and think of F x;R (·) as a random field, as x ∈ T 2 varies randomly uniformly on the torus. In what follows we will obtain a sequence of random fields g n : R 2 → R, that will converge in suitable sense to g, and we will denote their scaled version g n ω;R (·) := g n ω (·R), that will be compared to the scaled version of g  . Let S ′′ ⊆ S be a sequence of energy levels satisfying the assumptions of Theorem 2.6. Then there exists a sequence of Gaussian stationary random fields {g n } n∈S ′′ , converging in law to g as n → ∞, with the following property. For every R > 0, ǫ > 0 and η > 0, there exists n 0 = n 0 (R; η, ǫ) sufficiently large so that for all n ∈ S ′′ with n > n 0 and f n ∈ B n , there exists an event Ω ′ = Ω ′ (n; f n , R; η, ǫ) ⊆ Ω of high probability Pr(Ω ′ ) > 1 − ǫ and a measure preserving map τ : Ω ′ → T 2 so that meas(τ (Ω ′ )) > 1 − ǫ, and for all ω ∈ Ω ′ , one has Since, as it was mentioned above, Proposition 4.3 was proved 7 in [8], there is no need to reprove it in this manuscript. Once the reduction to the Gaussian random field was performed within Proposition 4.3, replacing g n (·) with Berry's g(·) in (4.19) is completely standard. That is, it is possible to couple g n (·) with g(·) so that g n ω;R − g ω;R C 1 ([−1,1] 2 ) is arbitrarily small for n sufficiently large, see e.g. [33,Lemma 4]. Together with (4.19) and the triangle inequality it yields the following corollary. Then for every R > 0, ǫ > 0 and η > 0, there exists n 0 = n 0 (R; η, ǫ) sufficiently large so that for all n ∈ S ′′ with n > n 0 and f n ∈ B n , there exists an event Ω ′ = Ω ′ (n; f n , R; η, ǫ) ⊆ Ω of high probability Pr(Ω ′ ) > 1 − ǫ and a measure preserving map τ : Ω ′ → T 2 so that meas(τ (Ω ′ )) > 1 − ǫ, and for all ω ∈ Ω ′ , one has Alternatively to working with g(·), one could, in principle, work directly with g n (·), by proving an analogue of Lemma 4.5 below, applicable for g n (·) with n large, a direction we abandon. Corollary 4.4 naturally gives rise to the comparison to the defect variance of the random waves g(·). Note that, for our purposes of comparing the defect of the toral eigenfunctions to that of the random g R , the C 1 -estimate in (4.20) is too strong, and we could easily settle for an L ∞ -estimate. Recall that H(·) is the sign function (1.6), and let (4.21) X R = X ω,R := 1 πR 2

B(R)
H(g(x))dx be the (random) defect of g(·) restricted to the ball B(R) ⊆ R 2 . It is obvious that the expectation E[X R ] = 0 vanishes, whereas the following easy, most likely sub-optimal, result asserts that so does its variance, asymptotically as R → ∞.
Lemma 4.5. As R → ∞, the defect variance of g(·) restricted to B(R) is vanishing: Proof. We use the definition (4.21) of the defect, and invert the integration order to write arcsin(J 0 (|x − y|))dxdy, 7 In [8] a more general situation was considered, when the equidistribution assumption on the lattice points was lifted, whence the limit random field was varying, depending on their angular distribution, rather than the sole Berry's random waves limit field g(·).

(4.24)
We bound the contribution of the former range trivially as (4.25) whereas we use the standard asymptotics [1, formula (9.2.1)] for the Bessel J 0 function for |x−y| > 1: to bound the contribution of the latter range as The statement of Lemma 4.5 finally follows upon substituting (4.25) and (4.26) into (4.24).
where χ [−η,η] is the characteristic function of the interval [−η, η] ⊆ R. Since, for every x ∈ R 2 , g(x) is a standard Gaussian random variable, taking the expectation of both sides of (4.28) easily yields with the constant involved in the 'O'-notation absolute. Now, we have and, in light of (4.29), the conclusion of Lemma 4.7 follows from Markov's inequality.
After all the preparatory results of §4.3, we are finally in a position to prove the principal derandomization result.
4.4. Spatial defect distribution: Proof of Proposition 4.1 via Bourgain's de-randomization. We start with the following elementary lemma in probability theory, that is a criterion for the variance vanishing of bounded random variables, whose proof is thereupon conveniently omitted. Proof of Proposition 4.1. We are going to use Lemma 4.8 as a criterion for the variance vanishing, upon both exploiting the defect variance for Berry's random waves (Lemma 4.5), and also when proving the same for the toral eigenfunctions; note that the prescribed rate (4.22) is "lost" during this process for the latter. Let ǫ, δ > 0 be given. First, we invoke Lemma 4.7 on δ/4 in place of δ, and ǫ/2 in place of ǫ, to obtain a number η = η(ǫ/2, δ/4) sufficiently small so that for all R > 0, (4.30) Pr(Ω 1 (R, η, δ/4)) < ǫ/2.
Further, let w 1 , w 2 , w 3 ∈ R 2 denote elements corresponding to the three third roots of unity. Using that e(t) + e(−t) = 2 cos(t), and pairing off antipodal points (i.e. v i = −v j ) define the completely flat function Further, g m (x) := g(mx) is a Laplace eigenfunction on T with eigenvalue 4π 2 m 2 (also completely flat if m is chosen to be a prime that is inert in Z[e 2πi/3 ], and the following proposition asserts that the total defect of g does not vanish.  A plot of H(g(x 1 , x 2 )) is shown in Figure 1. Since g is invariant under translation by L, unless the integral over the fundamental domain of L is exactly zero, we will get growth, of order R 2 in either the positive or the negative direction, when integrating over squares, say centred at (R/2, R/2) and with sides length R growing. The numerics in Table 1 indicates that there is negative growth. These numerics can be made rigorous by bounding the gradient from above: this way we can ensure that the function does not change sign in most small disks. The following lemma, whose proof is obvious, introduces a stability notion, related to the one in section 4.3. Proof of Proposition 5.1. Recall that the lattice L is spanned by u 1 = (1, 1/ √ 3) and u 2 = (0, 2/ √ 3). The rhombus spanned by u 1 , u 2 is a fundamental domain of L, as well as a fundamental domain for T . As it is more convenient to tile with rectangles rather than with rhombi we will prefer to evaluate the signed area on a rectangular fundamental domain, and show that the defect integral over the rectangle R, having corners at (0, 0), (1, 0), (0, 2/ √ 3), (1, 2/ √ 3), easily seen to be a fundamental domain of T , is non-zero.
For some integer N > 0 we tile R by N 2 rectangles (modulo R) centred at for 0 ≤ j, k < N; each such rectangle can be covered with a disk of radius r = 7/12/N. If the inequality |g(h j,k )| > 12πr > r · M, with M as in (5.3) is satisfied (using a factor of two safety margin), the corresponding rectangle centred at h j,k is said to be "stable", whence g(·) has constant sign on the whole rectangle by Lemma 5.2; otherwise it is said to be "unstable". Depending on the sign of g(h j,k ), we call the corresponding stable rectangle "positively stable" or "negatively stable". For N = 80 one finds 2099 positively stable rectangles, 3299 negatively stable, and 1002 unstable ones. As 3299 − 2099 = 1200 > 1002, we conclude that the defect (5.2) is nonzero (and in fact negative). Both assertions of Proposition 5.1 now follow: the first assertion follows from the presented numerical calculation, whereas the second one is an immediate consequence of the first assertion upon tiling B x (s) with π(ms) 2 /(2/ √ 3)+O(ms) copies of fundamental domains associated with the lattice 1 m L (note that the boundary of B x (s) can be covered with O(ms) tiles.) One can obtain more precise estimates on c in (5.2), by increasing N, and thus decreasing the mesh size: for example, for N = 500, the corresponding counts are respectively 96639, 147207, and 6154.

5.2.
Defect stability w.r.t. perturbations of g. For later use we show that a small perturbation of g only changes the defect by a small amount. For convenience we work in the rescaled region where the eigenvalues are normalized to 4π 2 , hence we should consider the defect over balls of radius R (or squares of sides R) with R growing. We start by showing that simultaneous vanishing of both g and its gradient ∇g is impossible. Proof. The linear map R 3 → R 2 , given by (a 1 , a 2 , a 3 ) → 3 i=1 a i w i with w i as in (5.1), clearly has full range, hence a one dimensional kernel, spanned by (1, 1, 1). In particular, if 3 i=1 a i w i = 0, then a 1 = a 2 = a 3 = C for some C. Therefore, ∇g(x) = 0 implies that cos(2πw 1 · x) = cos(2πw 2 · x) = cos(2πw 3 ·x) = C for some C. Further, g(x) = 0 implies that 0 = 3 i=1 cos(2πw i ·x) = 3C, and thus C = 0 for any point where g and ∇g both vanish. In particular, we find that 2πw i · x = ±π/2 + 2πk i for k i ∈ Z. On the other hand, as 3 i=1 w i = 0, we find, on multiplying by 2/π that 0 ≡ ±1 + ±1 + ±1 mod 4 which is impossible since the right hand side is odd no matter what signs are chosen.
In light of Lemma 5.3 and the compactness of T , it follows that the gradient of g is uniformly bounded below on the zero set of g(·): Corollary 5.4. There exist C > 0 such that |∇g(x)| ≥ C for all x ∈ Z 1 = g −1 (0).
It is now straightforward to prove stability of the defect of g w.r.t. perturbations. Given R ≥ 1 and a continuous function f ∈ C(R 2 ), define Proof. It is sufficient to show that the measure of the set {x ∈ T : |g(x)| ≤ ǫ} is O(ǫ), for all sufficiently small ǫ, as we can then tile B x (R) with ∼ R 2 copies of the fundamental domain. Now, there exist some open neighborhood of Z 1 = g −1 (0), outside of which |g(x)| is uniformly bounded away from zero (say, using compactness of the closed complement). In other words, if |g(x)| is small then we must have d(x, Z 1 ) small, where d(x, Z 1 ) denotes the distance between x and the zero set Z 1 . Further, all x for which d(x, Z 1 ) is sufficently small is contained in some small tubular neighbourhood of Z 1 . The lower bound on the gradient of Corollary 5.4 implies that |g(x)| ≫ d(x, Z 1 ) + O(d(x, Z 1 ) 2 ), and hence the measure of the set of x for which |g(x)| < ǫ is ≪ ǫ.

5.3.
Approximating g on the standard torus T 2 = R 2 /Z 2 : proof of Theorem 1.5. We next show that a perturbed variant of the hexagonal lattice construction can be translated to the square torus. We begin by showing that the set of Gaussian integers, scaled to have norm one, can very well approximate third roots of unity.
Proof of Theorem 1.5. We claim that the statement of Theorem 1.5 holds, with S ′′′ prescribed by (5.5), satisfying, in particular, the statement (5.7) of Proposition 5.6. To construct eigenfunctions on T = R 2 /Z 2 having large defect it is convenient to rescale T so that the eigenvalue equals 4π 2 , and correspondingly the torus must be rescaled so that the fundamental domain is a square with sides n 1/2 (where λ = 4π 2 n denotes the unscaled eigenvalue.) Given n = a 2 + b 2 ∈ S ′′′ with b 2 − 3a 2 = 1 define the unit vectors w i := z i |z i | ∈ R 2 , i = 1, 2, 3, with z i as in (5.6), and the Laplace eigenfunction G, on the re-scaled torus R 2 /( √ nZ 2 ), by A simple calculation shows that G is a Laplace eigenfunction, with eigenvalue 4π 2 , and that, with w i as in (5.1), the asymptotic approximation (5.8) reads Hence, for any x ∈ R 2 , we have |g(x) − G(x)| ≪ |x|/n 1/2 .
In particular, for |x| = o(n 1/2 ), we have G(x) = g(x) + o(1), and thus, if R = o(n 1/2 ) grows with n we find, thanks to Lemma 5.5, that Y G,R (x) = Y g,R (x) + o(1) = C + o(1) for C := c · √ 3/2 < 0. In the macroscopic regime, i.e. when R is of size n 1/2 , we similarly find that for |x| ≪ ǫn 1/2 , Y G,R (x) = Y g,R (x) + O(ǫ) = C + O(ǫ). Thus, if for n ∈ S ′′′ we construct G as described above and define f n (x) := G( √ nx), we obtain an eigenfunction on T 2 , with eigenvalue 4π 2 n, and find that the defect integral over B x (s) (keeping in mind that s = R/ √ n when we undo the scaling) is bounded away from zero for |x| < ǫ; hence the variance is bounded from below, and the proof is concluded.