Asymptotic nodal length and log-integrability of toral eigenfunctions
AASYMPTOTIC NODAL LENGTH AND LOG-INTEGRABILITY OFTORAL EIGENFUNCTIONS
ANDREA SARTORI
Abstract.
We study the zero set of “generic” Laplace eigenfunctions on the standardtwo dimensional flat torus T = R / Z . We find the asymptotic nodal length in any ballof radius larger than the Planck-scale. In particular, we prove that the nodal set equidis-tributes on T . Moreover, we show that the nodal set equidistributes at Planck-scalearound almost every point for some eigenfunctions and give examples of eigenfunctionswhose nodal set fails to equidistribute at Planck-scale. The proof is based on Bour-gain’s de-randomisation and the main new ingredient is the integrability of arbitrarilylarge powers of the logarithm of “generic ”eigenfunctions, based on the work of Nazarov[27, 28], and some arithmetic ingredients called semi-correlations. Introduction
Nodal length of Laplace eigenfunctions and the Random Wave Model.
Given a compact C ∞ -smooth surface Riemannian manifold ( M, g ) without boundary, let∆ g be the Laplace-Beltrami operator on M . Then, there exists an orthonormal basis for L ( M, dvol ) consisting of eigenfunctions { f λ i } ∆ g f λ i + λ i f λ i = 0with 0 = λ < λ ≤ ... repeated accordingly to multiplicity, and λ i → ∞ . Given aneigenfunction f λ with eigenvalue λ , its nodal set Z ( f λ ) = { x ∈ M : f λ ( x ) = 0 } is asmooth 1-dimensional sub-manifold [10] and Yau [33] conjectured √ λ (cid:46) M L ( f λ ) (cid:46) M √ λ, where L ( f λ ) = H{ x ∈ M ; f λ ( x ) = 0 } is the nodal length with respect to the Hausdorffmeasure and A (cid:46) M B means that there exists some constant C = C ( M ) > A ≤ CB . Donnelly and Fefferman [11] showed that Yau’s conjecture holds for any real-analytic manifold. Recently, Logunov and Malinnikova [20, 21, 22] proved the lower-boundin the smooth case and gave a polynomial upper-bound.Berry [2, 3] conjectured that “generic”Laplace eigenfunctions, in balls of radius slightlylarger than O ( λ − / ), the Planck-Scale , should behave as isotropic Gaussian field F withcovariance function E [ F ( x ) F ( y )] = J ( | x − y | )2 π , where J ( · ) is the 0-th Bessel function. This is known as the Random Wave Model (RWM).The RWM suggests that, for “generic”eigenfunctions, the nodal length should satisfy thestronger estimate L ( f λ ) = c √ λ (1 + o λ →∞ (1)) , (1.1) a r X i v : . [ m a t h . SP ] J a n ODAL LENGTH OF TORAL EIGENFUNCTIONS 2 for some absolute constant c >
0. Moreover, the nodal set should asymptotic equidis-tribute on every ball of radius slightly larger than Planck-scale, that is, we expect L ( f λ , B ) = H{ x ∈ B : f λ ( x ) = 0 } = c Vol( B ) √ λ (1 + o λ →∞ (1)) , (1.2)for any ball B = B ( r ) is any ball of radius r = r ( λ ) > r · λ / → ∞ . Inparticular, we expect the nodal set to equidistribute on M , see also [34, Chapter 13].We study Laplace eigenfunctions on the standard two dimensional torus T = R / Z and show that (1.2) holds for “generic”eigenfunctions provided that r > λ − / ε ; this, inparticular, implies (1.1) and the equidistribution of the nodal length on T . To the bestof the author knowledge, no other, non-trivial (e.g. f λ ( x ) = cos( a · x ) with | a | = λ ),examples of (1.2) or even (1.1) are known. One of the main ingredients in the proof isa de-randomisation technique pioneered by Bourgain [7] and its subsequent developmentby Buckley-Wigman [8].In [7, 8], it is shown that “generic”toral eigenfunctions satisfy the RWM, see Proposition2.2 below. The authors then deduced the asymptotic nodal domains count, the numberof connected components of T \Z ( f λ ), for the said eigenfunctions. In our work, we haveto face the extra difficulty posed by the possible concentration of nodal length in a smallproportion of space, that is the existence of few Planck-scaled balls which significantlycontribute to (1.1). This problem is not present for the nodal domains count thanks tothe Faber-Krahn inequality [25]. To overcome the said problem, we show that arbitrarilylarge powers of the logarithm of “generic ”eigenfunctions are integrable using the work ofNazarov [27, 28].As a consequence of our method, we are also able to study the distribution, at Planck-scale, of the nodal length for “generic ”eigenfunctions. We show that, if r · λ / → ∞ arbitrarily slowly then (1.2) is almost always true, that is, as we average x ∈ T , bearingin mind the rescaling factor for the nodal length, we haveVar( r L ( f, B ( x, r ))) = (cid:90) T (cid:0) r L ( f, B ( x, r )) − crλ / (cid:1) dx → r · λ / → ∞ . (1.3)However, we also give example where (1.3) fails and thus the is no equidistribution ofthe nodal length at Planck-scale. These results appears to be the first to address thedistribution of the nodal length of Laplace eigenfunctions at small scale and it comple-ments similar founding for the Planck-scale mass-distribution of toral eigenfunctions byLester-Rudnick [19], Granville-Wigman [14] and Wigman-Yesha [32], see also Humphries[15] for similar work on the modular surface.1.2. Toral eigenfunctions.
A Laplace eigenfunction on T with eigenvalue − πλ (wewill simply say eigenvalue λ from now on) can be written explicitly as a Fourier sum f λ ( x ) = f ( x ) = (cid:88) ξ ∈ Z | ξ | = λ a ξ e ( (cid:104) ξ, x (cid:105) ) . (1.4)where e ( · ) = e (2 πi · ) and the a ξ ’s are complex numbers satisfying a ξ = a − ξ so that f isreal valued. The eigenvalues are then integers representable as the sum of two squares λ ∈ S = { λ : λ = (cid:3) + (cid:3) } and have multiplicity N = N ( λ ) = |{ ξ ∈ Z : | ξ | = λ }| givenby the number lattice points on the circle of radius λ / . ODAL LENGTH OF TORAL EIGENFUNCTIONS 3
We normalise f as in (1.4) so that || f || L ( T ) = (cid:88) ξ | a ξ | = 1 , (1.5)and say that f is flat if, for every ε >
0, we have | a ξ | (cid:46) N − ε , (1.6)for all | ξ | = λ , where the constant implied in the notation is independent of N . Flatness makes precise the term “generic”toral eigenfunctions, see also discussion in [8]. Further-more, to f we associate the pseudo-spectral measure supported on the circle S ⊂ R and,for an integer k ≥
0, its Fourier coefficients as µ f = (cid:88) ξ | a ξ | δ ξ/ √ λ ˆ µ f ( k ) = (cid:90) S z k dµ f ( z ) , (1.7)where δ ξ is the Dirac distribution at the point ξ . These will be fundamental in describingthe asymptotic behaviour of L ( f ).Finally, since the set of probability measure on S , equipped with the weak (cid:63) topology,is compact, upon passing to a subsequence, form now on we always assume that µ f , asin (1.7), weak (cid:63) converges to some probability measure µ , as N → ∞ . Moreover, to avoiddegeneracies, we assume the support of µ is not contained in a line. For a study of theweak (cid:63) of µ f in the case | a ξ | = 1, see [18, 30].1.3. Statement of the main results.
Before stating the main results of the article, weintroduce the following (standard) piece of notation: a subsequence S (cid:48) ⊂ S is of densityone if lim X →∞ |{ λ ∈ S (cid:48) : λ ≤ X }||{ λ ∈ S (cid:48) : λ ≤ X }| = 1. Importantly, every subsequence S (cid:48) ⊂ S , discussed in thisscript, can be explicitly described by some arithmetic conditions, see Section 2.2 below.We are now ready to state our main results: Theorem 1.1.
Let ε > and µ be as in section 1.2. There exists a density one subse-quence of λ ∈ S and some explicit constant c = c (ˆ µ (2)) > such that L ( f, B ) = c Vol( B ) λ / (1 + o λ →∞ (1)) , uniformly for all flat f , satisfying µ f → µ , and all balls B ⊂ T of radius r ≥ λ − / ε ,where the limit is take along said density one subsequence. In particular, L ( f ) = c λ / (1 + o λ →∞ (1)) . Remark . Under the additional symmetry assumption that Im(ˆ µ (2)) = 0, then c inTheorem 1.1 can be given explicitly in terms of elliptic integrals as c ( µ ) = (1 − ˆ µ (2) )2 / π (cid:90) π − ˆ µ (2) cos(2 θ )) / dθ. To state the next theorem, given R ≥
1, we write F x,R ( y ) = F x = f (cid:18) x + Ry √ λ (cid:19) (1.8)for the restriction of f to the box B ( x, R/ √ λ ) and y ∈ [ − / , / . We also write L ( F x ) = L ( F x , B (1)) = H{ y ∈ B (1) = [ − / , / : F x ( y ) = 0 } . We will prove thefollowing: ODAL LENGTH OF TORAL EIGENFUNCTIONS 4
Theorem 1.3.
Let F x be as in (1.8) and µ be as in section 1.2. There exists a densityone subsequence of λ ∈ S such that the following holds: suppose that µ does not have anyatoms then there exists some constant c = c (ˆ µ (2)) such that Var ( L ( F x )) = c (cid:0) R + o R →∞ ( R ) (cid:1) (1 + o λ →∞ (1)) , uniformly for all flat f , satisfying µ f → µ , and the order of limits is λ → ∞ followed by R → ∞ . Moreover, if ˆ µ (2) = 0 then c = 0 .Remark . We use the assumption that µ does not have atoms to simplify some cal-culations in the study of the nodal length variance of the (centred) Gaussian field withspectral measure µ . It is very plausible that this assumption can be removed at the costof a more technical computation.In particular, we have the following corollary about the equidistribution of the lengthat Planck-scale around almost every point: Corollary 1.5.
Let (cid:15) > , then, under the assumptions of Theorem 1.3 and ˆ µ (2) = 0 , wehave lim R →∞ lim λ →∞ Vol (cid:18)(cid:26) x ∈ T : (cid:12)(cid:12)(cid:12)(cid:12) L ( F x ) R − √ (cid:12)(cid:12)(cid:12)(cid:12) > (cid:15) (cid:27)(cid:19) −→ . Example 1.6.
An important class of examples is given by “Bourgain’s eigenfunctions”: f ( x ) = 1 √ N (cid:88) | ξ | = E e ( (cid:104) x, ξ (cid:105) ) , note that these eigenfunctions are flat. Moreover, thanks to the equidistribution of latticepoints on S for a density one subset of λ ∈ S [12, 16], we may assume that limitingmeasure µ is the Lebesgue measure on the unit circle. Thus, Theorem 1.1 gives L ( f ) = 12 √ √ λ (1 + o (1)) . Furthermore, substituting Proposition 5.2 below with the result of Berry [4, Equation 28]evaluating the asymptotic variance of the nodal length of a Gaussian field with spectralmeasure µ , we obtain the precise asymptotic behaviourlim λ →∞ Var( L ( F x )) = log R π (1 + o R →∞ (1)) , where F x ( · ) = f ( x + Ry/ √ E ) and f is Bourgain.On the other hand, consider the following slightly modified Bourgain’s eigenfunctions:take all the coefficients supported on points in the third quadrant to be zero and allother coefficients to be 2 / √ N . Let us call such eigenfunctions ˜ f and observe that ˜ f isalso flat. Then, for a density one of eigenvalues, the limiting measure ˜ µ is the Lebesguemeasure supported on S minus the third quadrant. Thus, ˆ µ (2) (cid:54) = 0 and ˜ µ has no atoms.Computing the explicit value of c (˜ µ ) in Theorem 1.3 via equation (A.35) below, we have R (cid:46) Var( L ( ˜ F x )) . Hence, the nodal length does not equidistribute at Plank scale.Finally, the main new ingredient in the proof, which allows us to overcome the nodallength concentration, is the log-integrability of f , that is ODAL LENGTH OF TORAL EIGENFUNCTIONS 5
Proposition 1.7.
Let f be as in (1.4) and p ≥ be fixed. Then there exists a densityone subset of λ ∈ S such that (cid:90) T | log | f ( x ) || p dx ≤ C, for some absolute constant C > . In particular, as we will see in section 3.2, Proposition 1.7 imply that arbitrarily largepowers of the doubling index of “generic”toral eigenfunctions are integrable.1.4.
Outline of the proof.
In this outline, we always assume that f as in (1.4) if flat,let F x be as in (1.8), and for, simplicity, we only sketch the case B = T of Theorem 1.1.By Bourgain’s de-randomisation, Proposition 2.2 below, there exists a coupling such that || F x − F µ || C [ − / , / d −→ λ → ∞ (1.9)where the limit is taken along a density one subsequence of λ ∈ S and the convergenceis in distribution. From (1.9) and the stability of the nodal set, Lemma 4.2 below, wededuce that L ( F x ) d → L ( F µ ) λ → ∞ . (1.10)Now, we wish to extend the convergence in (1.10) to convergence of moments. To thisend, the estimates on the nodal length of Laplace eigenfunctions (on analytic manifolds)by Donnelly and Fefferman, Lemma 3.2 below, give L ( F x ) (cid:46) log sup B ( x, R √ E ) | f | sup B ( x, R √ E ) | f | , where B ( x, r ) is the ball centred at x ∈ T of radius r >
0. Therefore, given p ≥ (cid:90) x ∈ T L ( F x ) p dx ≤ C (1.11)for some absolute constant C >
0. Thus, from (1.10) and (1.11), we have (cid:90) T L ( F x ) p dx −→ E [ L ( F µ ) p ] λ → ∞ (1.12)along a density one subsequence of λ ∈ S . Finally moments of L ( F µ ) can be computedusing the Kac-Rice formula, Propositions 5.1 and 5.2 below, concluding the proof ofTheorems 1.1 and 1.3.The proof of Proposition 1.7 relies on a Theorem of Nazarov [27, 28], see Theorem2.6 below, about log-integrability of Fourier series with spectrum supported on Λ( p )-systems, see section 2.3 for the relevant definitions. The recent work of Cammarota,Klurman and Wigman, Lemma 2.4 below, shows that the ξ = ( ξ , ξ )’s as in (1.4), or,more precisely, their projection on the first coordinate ξ , form a Λ( p ) system for a densityone subsequence of λ ∈ S . Hence, Proposition 1.7 follows from Theorem 2.6 and Lemma2.4. ODAL LENGTH OF TORAL EIGENFUNCTIONS 6 Preliminaries
Probability background.
Gaussian fields.
We briefly collect some definitionsabout Gaussian fields (on R ). A (real-valued) Gaussian field F is a continuous map F : R × Ω → R for some probability space Ω, equipped with a probability measure P ( · ), such that all finite dimensional distributions ( F ( x , · ) , ...F ( x n , · )) are multivariateGaussian. F is centred if E [ F ] ≡ stationary if its law is invariant under translations x → x + τ for τ ∈ R . The covariance function of F is E [ F ( x ) · F ( y )] = E [ F ( x − y ) · F (0)] . Since the covariance is positive definite, by Bochner’s theorem, it is the Fourier transformof some measure µ on the R . So we have E [ F ( x ) F ( y )] = (cid:90) R e ( (cid:104) x − y, λ (cid:105) ) dµ ( λ ) . The measure µ is called the spectral measure of F and, since F is real-valued, satisfies µ ( − I ) = µ ( I ) for any (measurable) subset I ⊂ R , i.e., µ is a symmetric measure. ByKolmogorov theorem, µ fully determines F , so we may simply write F = F µ . Moreover,to simplify notation, given a parameter R >
1, we write F µ ( · ) := F µ ( R · ), that is everyGaussian field is scaled by a factor of R , compare with the definition of F x in (1.8). The L´evy–Prokhorov metric.
Here we define the L´evy–Prokhorov metric and statesome standard results which will be useful later. Let C s ( V ) be the space of s -times, s ≥ V , a compact set of R . Since C s ( V ) is a separable metric space, Prokhorov’s Theorem, see [5, Chapters 5 and 6],implies that P ( C s ( V )), the space of probability measures on C s ( V ), is metrizable via the L´evy–Prokhorov metric . This is defined as follows: for a subset B ⊂ C s ( V ), let denotedby B + ε the ε -neighbourhood of B , that is B + ε := { p ∈ S | ∃ q ∈ B, d ( p, q ) < ε } = (cid:91) p ∈ B B ε ( p ) , where B ε ( p ) is the open ball of radius ε centred at p . The L´evy–Prokhorov metric d P : P ( C s ( V )) × P ( C s ( V )) → [0 , + ∞ ) is defined for two probability measures µ and ν as: d P ( µ, ν ) := inf ε> { µ ( B ) ≤ ν ( B + ε ) + ε, ν ( B ) ≤ µ ( B + ε ) + ε ∀ B ∈ B ( S ) } . (2.1)We will also need the following result of uniform integrability [5, Theorem 3.5]. Lemma 2.1.
Let X n a sequence of random variables such that X n d → X , that is indistribution. Suppose that there exists some α > such that E [ | X n | α ] ≤ C < ∞ forsome C > , uniformly for all n ≥ . Then, E [ X n ] → E [ X ] n → ∞ . The push-forward measure
Given a ball B ⊂ T of radius r > F x as in (1.8), wedenote by Vol B the uniform measure on B , that is for a test-function g : B → R , we have (cid:90) g ( x ) d Vol B ( x ) := 1 πr (cid:90) B g ( x ) dx. Then, for an integer s ≥ F x induces a probability measure on P ( C s ([ − / , / )), viathe push-forward ( F x ) (cid:63) Vol B ( A ) = Vol B ( { x ∈ B : F x ( y ) ∈ A } ) , ODAL LENGTH OF TORAL EIGENFUNCTIONS 7 where A ⊂ C s ([ − / , / ) is a measurable subset. Similarly, given a (symmetric)probability measure µ on S , the push-forward of F µ defines a probability measure on P ( C s ([ − / , / )) which we denote by ( F µ ) (cid:63) P . To shorten notation, we simply write d P ( F Bx , F µ ) = d P (( F x ) (cid:63) Vol B , ( F µ ) (cid:63) P ) . Bourgain’s de-randomisation and correlations.
Bearing in mind the notationintroduced in the previous section, we have the following result [7, 8] and [31, Proposition4.5] for the small scales version:
Proposition 2.2.
Let
R > , ε > , f , F x be as (1.4) and (1.8) respectively and F µ bethe centred stationary Gaussian field with spectral measure µ , where µ is as in section 1.2.Then there exists a density one subsequence of λ ∈ S such that d p ( F Bx , F µ ) → λ → ∞ uniformly for all f flat, satisfying µ f → µ , and all balls B ⊂ T of radius r > λ − / ε ,where the convergence is with respect to the C ([ − / , / ) metric. The proof of Proposition 2.2 relays on the following result by Bombieri and Bourgain[6, Theorem 14]:
Lemma 2.3 ( Bombieri-Bourgain) . Let (cid:96) be a positive integer. Then, for a density onesubsequence of λ ∈ S , there exists no non-trivial solution to the linear equation ξ + ... + ξ (cid:96) = 0 | ξ | = λ. That is all solutions have the form ξ = − ξ ,..., ξ (cid:96) − = − ξ (cid:96) . We will need the following version of Lemma 2.3 for projections of the ξ onto the firstcoordinate [9, Theorem 1.3]: Lemma 2.4.
Let ξ = ( ξ , ξ ) ∈ Z and (cid:96) ≥ . Then, for a density one subsequence of λ ∈ S , there exists no non-trivial solution to the linear equation ξ + ... + ξ (cid:96) = 0 | ξ | = λ. Log-integrability and Λ( p ) -systems. We need to first introduce some definitions,given some g ∈ L ( T ), the spectrum of g , written Spec( g ), isSpec( g ) = { n ∈ Z : ˆ g ( n ) (cid:54) = 0 } , where ˆ g ( n ) = (cid:82) e ( n · x ) g ( x ) dx . Moreover, we say that a (possibly finite) set V = { n i } i ⊂ Z is a Λ( p ) system for some p ≥ g ∈ L ( T ) with Spec( g ) ⊂ V , there existssome ˜ C >
0, independent of g , such that || g || L p ≤ ˜ C || g || L . (2.2) Claim 2.5.
Let V = { n i } i ⊂ Z such that, for some even p ≥ , the only solutions to n i + n i + ... + n i p = 0 are trivial, that is n i = − n i ... . Then, V is a Λ( p ) -system and ˜ C ( p ) = c ( p ) is indepen-dent of V . ODAL LENGTH OF TORAL EIGENFUNCTIONS 8
Proof.
Let g ∈ L ( T ) with Spec( g ) ⊂ V , then we may write the Fourier series of g as g ( x ) = (cid:88) i a i e ( n i · x ) , for some a i ∈ C and, normalising g , we may also assume that || g || L = (cid:88) i | a i | = 1 . Now, expanding the p -power of g , we have || g || pL p = (cid:88) i ,...,i p a i a i ...a i p (cid:90) e ( (cid:104) n i − n i + ... − n i p , x (cid:105) ) dx and, using the assumption and the orthogonality of the exponentials, we deduce || g || pL p = ( (cid:88) i | a i | ) p/ + ( (cid:88) i | a i | )( (cid:88) i a i ) p/ − + .... Since (cid:80) | a i | ≤ ( (cid:80) | a i | ) , we finally have || g || pL p ≤ c ( p ) , for some c ( p ) independent of g and of V , as required. (cid:3) Finally, given some V = { n i } i ⊂ Z , we let R ( V ) := sup r (cid:54) =0 |{ ( n i , n j ) ∈ V : n i − n j = r }| D ( V ) := { n i − n j ∈ Z : i (cid:54) = j } With the above notation, we have the following theorem [28].
Theorem 2.6 (Nazarov) . Let ε > , V ⊂ Z such that R ( V ) < ∞ and D ( V ) is a Λ( p (cid:48) ) system for some p (cid:48) > , moreover let ˜ C = ˜ C ( p (cid:48) ) be as in (2.2) . Then there exists someconstant C = C ( ˜ C, ε, R ( V )) > such that, uniformly for all g ∈ L ( T ) with spectrumcontained in V and normalised so that || g || L ( T ) = 1 , we have (cid:90) T | log | g ( x ) || p (cid:48) − ε dx ≤ C. For the sake of completeness, we provide a proof of Theorem 2.6 in Appendix B.3.
Log-integrability and moments of L ( F x )In this section, we will prove Proposition 1.7 and as a consequence we deduce thefollowing result: Proposition 3.1.
Let
R > , F x be as in (1.8) and p ≥ be fixed. Then there exists adensity one subset of λ ∈ S such that (cid:90) T L ( F x ) p dx (cid:46) R p . ODAL LENGTH OF TORAL EIGENFUNCTIONS 9
Log-integrability of toral eigenfunctions.
In this section, we prove Proposition1.7, the proof follows directly combining Lemma 2.4 and Theorem 2.6
Proof of Proposition 1.7.
Let p ≥ (cid:96) = 4 p + 2. Then Theorem 2.4, viaClaim 2.5, implies that there exists a density one subset of λ ∈ S such that E λ = { ξ : | ξ | = λ } , where ξ = ( ξ , ξ ), is a Λ(2 (cid:96) )-system. In particular, bearing in mind that if ξ ∈ E λ then − ξ ∈ E λ , D ( E λ ) is a Λ( (cid:96) )-system and R ( E λ ) = 3. Hence, writing (cid:90) T | log | f λ ( x ) || p dx = (cid:90) T (cid:90) T | log | f λ ( x , x ) || p dx dx , Proposition 1.7 follows from Theorem 2.6 applied to V = E λ , (cid:15) = 1 and p (cid:48) = (cid:96) = 4 p +2. (cid:3) Proof of Proposition 3.1.
In this section given a ball B ⊂ R n and r >
0, we write rB for the concentric ball of r -times the radius. To conclude the proof of Proposition 3.1,we need the following lemma, see [23, Lemma 2.6.1] and [11, Proposition 6.7]: Lemma 3.2.
Let B ⊂ R be the unit box, suppose that h : 3 B → R is an harmonicfunction, then V (cid:18) h, B (cid:19) (cid:46) log sup B | h | sup B | h | , where V (cid:0) h, B (cid:1) = H ( { x ∈ − B ; h ( x ) = 0 } ) . We are finally ready to prove Proposition 3.1:
Proof of Proposition 3.1.
First, we apply Lemma 3.2 to h ( y, t ) = F x (2 y ) e πRt : [ − , → R and B = [ − , to see that V ( h, / (cid:46) log sup B | h | sup B | h | . (3.1)Moreover, since L ( F x ) (cid:46) H ( h − (0)), bearing in mind the rescaling factor for the nodallength, we have L ( F x ( · )) (cid:46) L ( F x (2 · )) (cid:46) V ( h, / . Furthermore, by elliptic estimates [13, Page 332], we also havesup B | h | (cid:46) || h || L (3 B ) (cid:46) e πR (cid:90) [ − , | F x ( y ) | dy. Thus, given p ≥ X + Y ) p (cid:46) p X p + Y p , (3.1) gives L ( F x ) p (cid:46) p R p + (cid:16) log || F x || L [6 B ] (cid:17) p + | log | f ( x ) || p , (3.2)where we have used the lower bound sup B | h | ≥ | f ( x ) | and slightly abused notation writing B = [ − , . Now, integrating with respect to x ∈ T , the first term on the right handside of (3.2) is acceptable, the last term in bounded by Proposition 1.7, thus it is enoughto bound the second term. To this end, we observe that (cid:90) T || F x || L [6 B ] = O (1) , ODAL LENGTH OF TORAL EIGENFUNCTIONS 10 therefore Chebyshev’s inequality gives (cid:90) T (cid:16) log || F x || L [6 B ] (cid:17) p dx = (cid:90) ∞ µ (cid:16)(cid:16) log || F x || L [6 B ] (cid:17) p > t (cid:17) dt = (cid:90) ∞ µ (cid:16) || F x || L [6 B ] > exp( t p ) (cid:17) dt (cid:46) (cid:90) ∞ exp( − t p ) dt = O (1) . This concludes the proof of the Proposition. (cid:3) Convergence of moments
The aim of this section is to prove the following Proposition:
Proposition 4.1.
Let
R > , ε > , f , F x be as (1.4) and (1.8) respectively and F µ bethe centred stationary Gaussian field with spectral measure µ , where µ is as in section 1.2.Then for every p ≥ there exists a density one subsequence of λ ∈ S such that B ) (cid:90) B L ( F x ) p dx → E [ L ( F µ ) p ] λ → ∞ , uniformly for all flat f , satisfying µ f → µ , and all balls B ⊂ T of radius r > λ − / ε . To prove Proposition 4.1, we first prove convergence in distribution and then concludeusing Proposition 3.1.4.1.
Convergence in distribution.
Here we prove the following Lemma:
Lemma 4.2.
Let
R > , ε > , f , F x be as (1.4) and (1.8) respectively and F µ be thecentred stationary Gaussian field with spectral measure µ , where µ is as in section 1.2.Then there exists a density one subsequence of λ ∈ S such that L ( F Bx ) d → L ( F µ ) λ → ∞ , where the convergence is in distribution, uniformly for all flat f , satisfying µ f → µ , andall balls B ⊂ T of radius r > λ − / ε . The proof of the Lemma rests upon the two results. The first result is a consequence ofthe stability of the zero set under C -perturbations as in [26], see also [24, Lemma 6.1]: Lemma 4.3.
Let C ∗ ( B ) = { g ∈ C ( B ) : | g | + |∇ g | > , } . Then L ( g, B ) = { x ∈ B : g ( x ) = 0 } is a continuous functional on C ∗ (2 B ) . The other result is the following well-known result of Bulinskya, see for example [26,Lemma 6].
Lemma 4.4 (Bulinskya’s lemma) . Let F = F ν , with ν an hermitian measure supported on S . If ν is not supported on a line, that is ( F, ∇ F ) is non-degenerate, then F ∈ C ∗ ( B (2)) almost surely, where C ∗ ( B (2)) Lemma (4.3) . We are finally ready to prove Lemma 4.2
Proof of Lemma 4.2.
Since the support of µ is not contained in a line, Lemma 4.4 impliesthat F µ ∈ C ∗ ( B (2)) almost surely. Moreover, up to changing the value of R , Theorem 2.2implies that d p ( F Bx , F µ ) → , with respect to the C ( B (2)) topology. Hence, Lemma 4.2 follows from the ContinuousMapping Theorem [5, Theorem 2.7], together with Lemma 4.3. (cid:3) ODAL LENGTH OF TORAL EIGENFUNCTIONS 11
Proof of Proposition 4.1.
We are finally ready to prove Proposition 4.1:
Proof of Proposition 4.1.
Let p ≥ p instead of p and the Cauchy-Schwartz inequality give1Vol( B ) (cid:90) B L ( F x ) p dx ≤ (cid:18)(cid:90) T L ( F x ) p (cid:19) / (cid:46) R p . Thus Proposition 4.1 follows from Lemma 4.2 via Lemma 2.1. (cid:3) Concluding the proofs of Theorems 1.1 and 1.3
Before concluding the proofs of Theorems 1.1 and 1.3, we need a few preliminary results.5.1.
Nodal length of Gaussian random fields.
The main results of this section arethe following two Propositions the proofs of which are postponed to Appendix A.
Proposition 5.1.
Let µ be a probability measure supported on S , but not supported ona line. Then there exist an explicit constant c = c (Re(ˆ µ (2)) , Im(ˆ µ (2)) > such that E [ L ( F µ )] = c · R. Moreover, if
Im(ˆ µ (2)) = 0 , then c = 1 − ˆ µ (2) / π (cid:90) π − ˆ µ (2) cos(2 θ )) / dθ. Proposition 5.2.
Let
R > , µ be a probability measure on S , with no atoms. Thenthere exist an explicit constant c = c (Re(ˆ µ (2)) , Im(ˆ µ (2)) > such that Var( L ( F µ )) = c R + o ( R ) . Moreover, if ˆ µ (2) = 0 , then c = 0 . Locality nodal length.
Here we prove the following standard lemma:
Lemma 5.3.
Let B ⊂ T be a ball, R ≥ , f and F x be as in (1.4) and (1.8) respectively.Then, for a density one subsequence of λ ∈ S , we have L ( f, B ) = λ / R (cid:90) B L ( F x ) dx + O ( R ) Proof.
Let us write B = B ( z, r ) = B ( r ) for some z ∈ T and r >
0, then we observe that λ / R (cid:90) B ( r − R/λ / ) L ( F x ) ≤ L ( f, B ) ≤ λ / R (cid:90) B ( r + R/λ / ) L ( F x ) . (5.1)Indeed, writing r (cid:48) = R/λ / , by definition of L ( · ) and Fubini, we have (cid:90) B L ( f, B ( x, r − r (cid:48) )) dx = (cid:90) B ( r − r (cid:48) ) (cid:90) B ( r ) B ( x,r (cid:48) ) ( y ) f − (0) ( y ) d H ( y ) dx. = (cid:90) B ( r ) f − (0) ( y ) Vol ( B ( y, r (cid:48) ) ∩ B ( R )) d H ( y ) , ODAL LENGTH OF TORAL EIGENFUNCTIONS 12
Thus (5.1) follows from rescaling L ( f, B ( x, r (cid:48) )) = Rλ − / L ( F x ), upon noticing B ( r − r (cid:48) ) ≤ Vol ( B ( · , r (cid:48) ) ∩ B ( R ))Vol B ( r (cid:48) ) ≤ B ( r + r (cid:48) ) . Finally, by Proposition 3.1 with p = 2 and the Cauchy-Schwartz inequality, for a densityone of λ ∈ S , we have (cid:18)(cid:90) B ( r ) − (cid:90) B ( r ± R/λ / ) (cid:19) L ( F x ) dx (cid:46) r R λ / . (5.2)Hence the Lemma follows from (5.1) and (5.2), bearing in mind that r ≤ (cid:3) Proof of Theorem 1.1.
We are finally ready to prove Theorem 1.1:
Proof of Theorem 1.1.
Let B be given then by Lemma 5.3 and Proposition 4.1 with p = 1,for a density one of λ ∈ S , we have L ( f, B ) = λ / Vol( B ) R · B ) (cid:90) B L ( F x ) dx + O ( R ) = c Vol( B ) λ / (1 + o λ →∞ (1)) . Hence, the first part of the Theorem follows from Proposition 5.1 and the second followsfrom the first taking B = [ − / , / . (cid:3) Proof of Theorem 1.3.
We are finally ready to prove Theorem 1.3:
Proof of Theorem 1.3.
The proof is similar to the proof of Theorem 1.1. Proposition 4.1,for a density one of λ ∈ S , givesVar( L ( F x )) → Var[ L ( F µ )] . Hence, the Theorem follows from Proposition 5.2. (cid:3)
Acknowledgements
The author would like to thank Peter Sarnak for asking the question about finding theasymptotic nodal length which gave rise to this work. We also thank Igor Wigman forthe many discussions and Alon Nishry for pointing out the work of Nazarov and readingthe first draft of the article. This work was supported by the Engineering and PhysicalSciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training inGeometry and Number Theory (The London School of Geometry and Number Theory),University College London.
Appendix A. Kac-Rice
A.1.
Proof of Proposition 5.1.
In this section we prove Proposition 5.1 as a standardapplication of the Kac-Rice formula [1, Theorem 6.2].
Proof of Proposition 5.1.
We write F = F µ . Since µ is not supported on a line, ( F, ∇ F )is non-degenerate, thus we apply the Kac-Rice formula [1, Theorem 6.2] to see that E [ L ( F µ , B )] = (cid:90) B E [ |∇ F ( y ) || F ( y ) = 0] φ F ( y ) (0) dy, (A.1)where φ F ( y ) is the density of the random variable F ( y ). since F ( y ) is Gaussian with meanzero and variance 1, φ F ( y ) (0) = 1 / √ π ; since F and ∇ F are independent (this can be ODAL LENGTH OF TORAL EIGENFUNCTIONS 13 seen directly differentiating E [ F ( x ) ] = 1), E [ |∇ F ( y ) || F ( y ) = 0] = E [ |∇ F ( y ) | ]; since F isstationary, we also have E [ |∇ F ( y ) | ] = E [ |∇ F (0) | ]. Thus, (A.1) simplifies to E [ L ( F µ )] = 1 √ π · E [ |∇ F (0) | ] = R √ π E [ R − |∇ F (0) | ] . (A.2)Now, we compute the covariance function of ∇ F . Writing ˆ µ (2) = α + iβ = (cid:82) cos(2 θ ) dµ ( e ( θ ))+ i (cid:82) sin(2 θ ) dµ ( e ( θ )) and using the relations cos(2 θ ) = 2 cos ( θ ) − − ( θ ) andsin(2 θ ) = 2 sin( θ ) cos( θ ), we have R − E [ ∂ x F ( x ) F ( y )] | x = y = (cid:90) R λ dµ ( λ ) = (cid:90) cos ( θ ) dµ ( e ( θ )) = 12 + α R − E [ ∂ x F ( x ) F ( y )] | x = y = (cid:90) R λ dµ ( λ ) = (cid:90) sin ( θ ) dµ ( e ( θ )) = 12 − α R − E [ ∂ x ∂ y F ( x ) F ( y )] | x = y = (cid:90) R λ λ dµ ( λ ) = (cid:90) cos( θ ) sin( θ ) dµ ( e ( θ )) = β . (A.3)Therefore, the covariance matrix of R − ∇ F is L = (cid:20) + α β β − α (cid:21) det( L ) = 14 (cid:0) − α − β (cid:1) . (A.4)Since R − ∇ F (0) is a bi-variate Gaussian with mean 0 and Covariance L , given in (A.4), E [ | R − ∇ F (0) | ] = 1 π (1 − α − β ) / (cid:90) R (cid:112) x + y exp (cid:18) − x (1 − α ) + y (1 + α ) − βxy (1 − α − β ) (cid:19) . Letting C := E [ | R − ∇ F | (2 π ) − / ] proves the first part of the proposition. Finally, as-suming that β = 0 we may simplify the integral by passing to polar coordinates: E [ | R − ∇ F | ] = 1 π (1 − α ) / (cid:90) π dθ (cid:90) ∞ r exp (cid:18) − r (1 − α ) (1 − α cos(2 θ )) (cid:19) dr. Substituting r = ( ηy ) / , where η = (1 − α cos θ ) − (1 − α ) we have E [ | R − ∇ F | ] = 12 π (1 − α ) / (cid:90) π η / dθ (cid:90) ∞ y − / e − y dy = 12 π Γ(1 + 1 / − α ) (cid:90) π − α cos(2 θ )) / dθ. (A.5)Since Γ(3 /
2) = √ π/
2, the second part of Proposition follows from (A.2) and (A.5). (cid:3)
A.2.
Proof of Proposition 5.2.
In this section we prove Proposition 5.2 essentiallyfollowing [17]. To ease the exposition, we divide the proof in a series of steps and write F = F µ and L ( F ) = L ( F, B (1)).
ODAL LENGTH OF TORAL EIGENFUNCTIONS 14
Step 1 . The Kac-Rice formula for the second moment . To apply the Kac-Rice for-mula for the second moment [1, Theorem 6.3], we need to check that the distribu-tions ( F ( y ) , ∇ F ( y )) and ( F ( x ) , F ( y )) are non-degenerate for all x, y ∈ [ − / , / .Since µ does not have atoms it is not supported on a line, thus ( F ( y ) , ∇ F ( y ))) is non-degenerate. Since F is stationary, the covariance of ( F ( x ) , F ( y )) is equal to the covarianceof ( F (0) , F ( w )) which is A ( w ) = (cid:20) r ( w ) r ( w ) 1 (cid:21) det A ( w ) = 1 − r ( w ) (A.6)Therefore, ( F (0) , F ( w )) is degenerate if and only if µ is supported on a line, which is notthe case by assumption on µ .Applying [1, Theorem 6.3] and bearing in mind that F is stationary, we have E [ L ( F ) ] = (cid:90) [ − / , / × [ − / , / φ ( F ( x ) ,F ( y )) (0 , E [ |∇ F ( x ) ||∇ F ( y ) || F ( x ) = 0 , F ( y ) = 0] dxdy = (cid:90) [ − / , / φ ( F (0) ,F ( w )) (0 , E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] dw (A.7) Step 1.1. Preliminary simplifications . The covariance of the 6 dimensional vector ( F (0) , F ( w ) , ∇ F (0) , ∇ F ( w )) is Σ( w ) := (cid:20) A ( w ) B ( w ) B ( w ) T C ( w ) (cid:21) where A ( w ) is given by (A.6) and B ( w ) := (cid:20) E [ F (0) ∇ F (0)] E [ F (0) ∇ F ( w )] E [ F ( w ) ∇ F (0)] E [ F ( w ) ∇ F ( w )] (cid:21) C ( w ) := (cid:20) E [ ∇ F (0) · ∇ F (0)] E [ ∇ F (0) · ∇ F ( w )] E [ ∇ F ( w ) · ∇ F (0)] E [ ∇ F ( w ) · ∇ F ( w )] (cid:21) . From (A.6) it also follows that φ ( F (0) ,F ( w )) (0 ,
0) = 12 π (cid:112) − r ( w ) . (A.8)bearing in mind that F and ∇ F are independent and using (A.3), we may simply B and C as B ( w ) = (cid:20) ∇ r ( w ) −∇ r ( w ) 0 (cid:21) C ( w ) = (cid:20) R · L − H ( w ) − H ( w ) R · L (cid:21) where H ( w ) is the Hessian of r ( w ) and L is as in (A.4). The covariance of the Gaussianvector ( ∇ F (0) , ∇ F ( w )) conditioned on F (0) = 0 and F ( w ) = 0 is thenΥ( ω ) = C − B T A − B = (cid:20) R · L R · L (cid:21) − − r (cid:20) ∇ r · ∇ r T r ∇ r T · ∇ rr ∇ r · ∇ r T ∇ r T · ∇ r (cid:21) . (A.9)To simplify notation, we define X = − − r ∇ r · ∇ r T Y = − H − − r r ∇ r T · ∇ r. (A.10) ODAL LENGTH OF TORAL EIGENFUNCTIONS 15
Remark
A.1 . Observe that X and Y are uniformly bounded. Indeed, the non-diagonalentries are dominated by the diagonal one by the Cauchy-Schwartz inequality. The latterare bounded since X has negative entries while Υ has positive diagonal entries.To compute (A.7) we will study φ ( F (0) ,F ( w )) (0 ,
0) and E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] separately. We begin with φ ( F (0) ,F ( w )) (0 , Step 2.
The singular set . We say that a point x ∈ T is singular if there exist a subsetΛ x ⊂ S of measure µ (Λ x ) > / (cid:104) Rλ, x (cid:105) ) > / (cid:104) Rλ, x (cid:105) ) < − / λ ∈ Λ x . From now on, we only consider the case cos( (cid:104) Rλ, x (cid:105) ) > / − / , / into O ( R ) squares of side O (1 / √ R ) and we call a square singular if it contains a singular point, we let S be theunion of all singular squares. Observe that, if y is a point in a singular square I x , then | cos( (cid:104) Rλ, x (cid:105) ) − cos( (cid:104) Rλ, y (cid:105) ) | ≤ / (cid:104) Rλ, y (cid:105) ) ≥ /
2. Thus, r ( y ) = (cid:90) Λ x cos( (cid:104) Rλ, y (cid:105) ) + (cid:90) S \ Λ x cos( (cid:104) Rλ, y (cid:105) ) ≥ · − ≥ . and so (cid:90) [ − / , / r ( w ) dw ≥ (cid:18) (cid:19) Vol( S ) . (A.11)Moreover, we can bound the contribution of a singular square, see [29, section 6.3], as (cid:90) I x (cid:112) − r ( w ) dw (cid:28) R − . (A.12)Since there are at most O ( R ) squares, using (A.11) and (A.12), we obtain (cid:90) S (cid:112) − r ( w ) dw (cid:28) R Vol( S ) · R − (cid:28) (cid:90) [ − / , / r ( w ) dw. (A.13) Step 3.
Asymptotic behaviour of E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0]. FollowingBerry [4], we have √ α = 1 √ π (cid:90) ∞ (1 − e − αt/ ) dtt / . So we can write E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0]= 12 π (cid:90) ∞ (cid:90) ∞ [ v (0 , − v ( t, − v (0 , s ) + v ( t, s )] dtds ( ts ) / (A.14)where, letting |∇ F (0) | = W and |∇ F ( w ) | = W , ODAL LENGTH OF TORAL EIGENFUNCTIONS 16 v ( t, s ) = E (cid:20) exp (cid:18) − || W || t − || W || s (cid:19)(cid:21) = 1 (cid:112) (2 π ) det Υ (cid:90) (cid:90) exp (cid:18) − || w || t − || w || s (cid:19) exp (cid:18) −
12 ( ω , ω )Υ − ( ω , ω ) T (cid:19) dw dw = det (cid:18)(cid:20) tI sI (cid:21) + Υ − (cid:19) − det Υ / = det( I + M ) − / where Υ is as in (A.9) and M = (cid:20) √ tI √ sI (cid:21) Υ (cid:20) √ tI √ sI (cid:21) . (A.15) Step 3.1. Computing v ( t, s ). Expanding M in (A.15), we write I + M = (cid:20) I + tL R I + sL R (cid:21) + (cid:20) tX √ tsY √ tsY sX (cid:21) where X , Y are as in (A.10) and L R = R L , moreoverdet( I + M ) − / = det (cid:18)(cid:20) I + tL R I + sL R (cid:21)(cid:19) − / det (cid:16) I + ˜ M (cid:17) − / where ˜ M := (cid:18)(cid:20) I + tL R I + sL R (cid:21)(cid:19) − (cid:20) tX √ tsY √ tsY sX (cid:21) . Since det( I + tL R ) = 1 + R t + ( R t ) (1 − α − β ) /
4, we havedet (cid:18)(cid:20) I + tL R I + sL R (cid:21)(cid:19) = [(1 + R t + ( R t ) (1 − α − β ) / ×× (1 + sR + ( R s ) (1 − α − β ) / h ( R t, R s ) and ( I + tL R ) − = 1 h ( R t, (cid:20) R t (1 − α ) − R t β − R t β R t (1 + α ) (cid:21) . (A.16)Letting ˜ X = X ( I + tL R ) − and ˜ Y = ( I + tL R ) − Y , and using the fact that for a blockdiagonal matrix, det (cid:20) A BC D (cid:21) = det( A ) det( D − CA − B ), we obtaindet ( I + M ) − / = 1 h ( R t, R s ) det (cid:16) I + ˜ M (cid:17) − / = 1 h ( R t, R s ) det( I + t ˜ X ) − / det( I + s ˜ X − √ st ˜ Y ( I + t ˜ X ) − √ ts ˜ Y ) . (A.17) ODAL LENGTH OF TORAL EIGENFUNCTIONS 17
Moreover, det( I + t ˜ X ) − / = 1 + O ( t ˜ X ) (A.18)and, using ( I + t ˜ X ) − = I − t ˜ X + O ( ˜ X ), we also havedet( I + s ˜ X − √ st ˜ Y ( I + t ˜ X ) − √ ts ˜ Y ) = det( I + s ˜ X − st ˜ Y − √ ts ˜ Y t ˜ X √ ts ˜ Y + O ( ˜ X ˜ Y ))= 1 + O ( st ˜ X, st ˜ Y ) . (A.19)Thus, putting (A.17), (A.18) and (A.19) together, we obtain v ( t, s ) = 1 h ( R t, R s ) + O ( st ˜ X, st ˜ Y ) . (A.20) Step 3.2. The integrals over s and t . By (A.16), for t small enough, t ˜ X = tX + O ( t X )and t ˜ Y = tY + O ( t Y ) so f ( t, s ) − f ( t, − f ( s,
0) + f (0 ,
0) = tsX + tsY + O ( t sX, st Y )for s, t small enough; for t large enough, t ˜ X = O ( X ), t ˜ Y = O ( Y ). Thus, inserting (A.20)into (A.14), and transforming the variables, we have E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] = R π (cid:18)(cid:90) ∞ (cid:18) − h ( t, (cid:19) dtt / (cid:19) + O ( X, Y ) . (A.21) Step 4.
Asymptotic behaviour of the Variance . Recalling the notation in Step 2 for thesingular set S and (A.7), we write E [ L ( F ) ] = (cid:18)(cid:90) [ − / , / \ S + (cid:90) S (cid:19) φ ( F (0) ,F ( w )) (0 , E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] dw. Thanks to Remark A.1, we have E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] = O (1); thus,using the expansion φ ( F (0) ,F ( w )) (0 ,
0) = 1 + O ( r ( w ) ) on [ − / , / \ S , we get E [ L ( F ) ] = 1 π (cid:90) [ − / , / \ S E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] dw + O (cid:32)(cid:90) S (cid:112) − r ( w ) dw (cid:33) + O (cid:18)(cid:90) [ − / , / \ S r ( w ) dw (cid:19) . (A.22)Using again Remark A.1 and (A.11) to extend the first integral in (A.22) to [ − / , / ,using (A.13) to bound the second integral and the Cauchy Schwartz inequality to boundthe third one, we obtain E [ L ( F ) ] = 1 π (cid:90) [ − / , / E [ |∇ F (0) ||∇ F ( w ) || F (0) = 0 , F ( w ) = 0] dw + O (cid:18)(cid:90) [ − / , / r ( w ) dw (cid:19) . (A.23)Thus, inserting (A.21) into (A.23), transforming the variables, and taking C as in Propo-sition 5.1, we have ODAL LENGTH OF TORAL EIGENFUNCTIONS 18
Var[ L ( F )] = R π (cid:18)(cid:90) ∞ (cid:18) − h ( t, (cid:19) dtt / (cid:19) − C R + O (cid:18)(cid:90) [ − / , / X ( w ) dw (cid:19) + O (cid:18)(cid:90) [ − / , / Y ( w ) dw (cid:19) + O (cid:18)(cid:90) [ − / , / r ( w ) dw (cid:19) . (A.24) Step 5.
Bounding moments and derivatives of the covariance . For k ≥ (cid:90) [ − / , / r ( w ) k dw = o R →∞ (1) (A.25)Indeed, we have (cid:90) [ − / , / r ( w ) k dw = (cid:90) S ... (cid:90) S (cid:90) [ − / , / e ( wR ( λ + ... + λ k )) dµ ( λ ) ...dµ ( λ k ) dw, (A.26)and the inner integral in (A.26) is (cid:90) [ − / , / e ( wR ( λ + ... + λ k )) dw (cid:28) J ( || R ( λ + ... + λ k )) |||| R ( λ + ... + λ k )) || where J ( · ) is the Bessel function of the first kind. Since J ( T ) (cid:28) T / for T large enoughand J ( T ) /T = O (1) for T small, the integral in (A.26) can be bounded as (cid:90) [ − / , / r ( w ) k dw (cid:28) (cid:90) || λ + ... + λ k ||≥ / √ R J ( || R ( λ + ... + λ k )) |||| R ( λ + ... + λ k )) || dµ ( λ ) ...dµ ( λ k )+ (cid:90) || λ + ... + λ k ||≤ / √ R J ( || R ( λ + ... + λ k )) |||| R ( λ + ... + λ k )) || dµ ( λ ) ...dµ ( λ k ) (cid:28) R − / + O (cid:18)(cid:90) || λ + ... + λ k ||≤ / √ R dµ ( λ ) ...dµ ( λ k ) (cid:19) . (A.27)Finally, since µ has no atoms, fixing λ , ..., λ k − , we have (cid:90) ... (cid:90) || λ + ... + λ k ||≤ / √ R dµ ( λ ) ...dµ ( λ k ) = o R →∞ (1) . (A.28)Thus, (A.25) follows from (A.27) and (A.28).Taking k = 4 and k = 5 in (A.25), we have (cid:90) [ − / , / r ( w ) dw = o (1) (cid:90) [ − / , / r ( w ) ( ∂ w i r ( w )) dw = o ( R ) (A.29)for i = 1 ,
2. Taking k = 2 we have (cid:90) [ − / , / ( ∂ w i r ( w )) dw (cid:28) R (cid:90) [ − / , / r ( w ) dw = o ( R ) (A.30)and for j = 1 , (cid:90) [ − / , / ∂ w i ∂ w j r ( w ) dw (cid:28) R (cid:90) [ − / , / r ( w ) dw = o ( R ) (A.31) ODAL LENGTH OF TORAL EIGENFUNCTIONS 19
Step 6.
Concluding the proof . Using the same argument as in Step 4 to control the singularsquares, and writing (1 − r ) − = 1 + O ( r ), we have (cid:90) [ − / , / X ( w ) dw (cid:28) max i (cid:90) [ − / , / ( ∂ w i r ( w )) dw ++ O (cid:18) max i (cid:90) [ − / , / r ( w ) ( ∂ w i r ( w )) dw (cid:19) + O (cid:18)(cid:90) [ − / , / r ( w ) dw (cid:19) (A.32)and (cid:90) [ − / , / Y ( w ) dw (cid:28) max i,j (cid:90) [ − / , / ( ∂ w i ∂ w j r ( w )) dw + O (cid:18) max i,j (cid:90) [ − / , / r ( w ) ( ∂ w i r ( w )) dw (cid:19) + O (cid:18)(cid:90) [ − / , / r ( w ) dw (cid:19) . (A.33)where i, j = 1 ,
2. Inserting (A.29), (A.30) and (A.31) in (A.32) and (A.33), we have (cid:90) [ − / , / X ( w ) dw = o (1) (cid:90) [ − / , / Y ( w ) dw = o (1) . (A.34)Finally, inserting (A.34) into (A.24), we obtainVar[ L ( F )] = R π (cid:18)(cid:90) ∞ (cid:18) − h ( t, (cid:19) dtt / (cid:19) − C R + o ( R ) (A.35)This proves the first part of the proposition with C = (1 / π )( (cid:82) (1 − /h ( t, dt/t / ) − C . If ˆ µ (2) = 0 we can use the formulas1 √ π (cid:90) ∞ (cid:18) − h ( t, (cid:19) dt = 1 √ π (cid:90) ∞
11 + t dt = 12 √ C = 12 √ Appendix B. Log-integrability
The content of this section follows [27, Chapter 3] and [28], we claim no originality andencourage the reader to see directly [27, 28].B.1.
Small values of Fourier series and Λ( p ) -systems. First, with the notation ofsection 2.3, we observe that Theorem 1.7 follows directly from the following result:
Theorem B.1 (Nazarov) . Let V ⊂ Z such that R ( V ) < ∞ and D ( V ) is a Λ( p (cid:48) ) systemfor some p (cid:48) > , moreover let ˜ C = ˜ C ( p (cid:48) ) be as in (2.2) . Then there exists some constant C = C ( ˜ C, ε, R ( V )) > such that uniformly for all g ∈ L ( T ) with spectrum contained in V , we have || g || L ≤ exp (cid:32) Cµ ( U ) p (cid:48) + ε (cid:33) (cid:90) U | g ( x ) | dx, for any positive measure set U ⊂ T , where µ ( · ) is the Lebesgue measure on T . We are now going to briefly show how Theorem B.1 implies Theorem 2.6:
ODAL LENGTH OF TORAL EIGENFUNCTIONS 20
Proof.
Let ε > (cid:90) T | log | g ( x ) || p (cid:48) − ε dx = (cid:90) ∞ µ (cid:16) | log | g ( x ) || p (cid:48) − ε > t (cid:17) dt ≤ (cid:90) ∞ µ (cid:16) log | g ( x ) | > t p (cid:48) + ε (cid:17) dt + (cid:90) ∞ µ (cid:16) log | g ( x ) | ≤ − t p (cid:48) + ε (cid:17) dt. (B.1)Since (cid:82) | g | dx = 1, Chebyshev’s inequality gives that the first term on the right hand sideof (B.1) is bounded by some constant C ( p (cid:48) , ε ) >
0. Thus, it is enough to show that (cid:90) ∞ µ (cid:16) log | g ( x ) | ≤ − t p (cid:48) + ε (cid:17) dt ≤ C ( ˜ C ( p (cid:48) ) , ε, R ( V )) . To this end, let 0 < δ < U δ = { x ∈ T : | g ( x ) | ≤ δ } , then, for some C = C ( ˜ C ( p (cid:48) ) , ε, R ( V )) >
0, Theorem B.1 gives1 ≤ exp (cid:32) Cµ ( U δ ) p (cid:48) + ε (cid:33) µ ( U δ ) δ ≤ exp (cid:32) Cµ ( U δ ) p (cid:48) + ε (cid:33) δ as µ ( U δ ) ≤
1, for some ε > C, c >
0, we have µ ( U δ ) ≤ C ( − log δ ) − p (cid:48) + cε Taking δ = exp( − t p (cid:48) + ε ) and choosing ε appropriately in terms of ε , we deduce that µ (cid:16) log | g ( x ) | ≤ − t p (cid:48) + ε (cid:17) ≤ C ( ˜ C ( p (cid:48) ) , (cid:15), R ( V )) t − − ε/ , which, once inserted in (B.1), implies the Theorem. (cid:3) Therefore, it will be enough to prove Theorem B.1.B.2.
Proof of Theorem B.1.
The main ingredient in the proof of Theorem B.1 is thefollowing Lemma, which we will prove in section B.3 below, see [27, Corollary 3.5] and[28].
Lemma B.2 (Spreading Lemma) . Let V = { n i } i ⊂ Z such that R ( V ) < ∞ and let U ⊂ T with µ ( U ) ≤ R ( V ) / (4( R ( V ) + 1) . Suppose that there exists some integer m ≥ such that µ ( U (cid:48) ) (cid:88) n i (cid:54) = n j (cid:12)(cid:12) ˆ U (cid:48) ( n i − n j ) (cid:12)(cid:12) ≤ m + 1 , for all subsets U (cid:48) ⊂ U of measure µ ( U (cid:48) ) ≥ µ ( U ) / . Then, there exists a set U ⊃ U suchthat(1) The measure of U satisfies µ ( U \ U ) ≥ µ ( U )4 m . (2) For all g ∈ L ( T ) with Spec( g ) ⊂ V , we ave (cid:90) U | g ( x ) | dx ≤ (cid:18) C m µ ( U ) (cid:19) m (cid:90) U | g ( x ) | dx, for some absolute constant C > . We will also need the following two claims:
ODAL LENGTH OF TORAL EIGENFUNCTIONS 21
Claim B.3.
Under the assumptions of Theorem B.1, the integer m > in Lemma B.2can be taken to be m = (cid:34) ˜ C R (Λ) µ ( U (cid:48) ) p (cid:35) =: (cid:34) Bµ ( U (cid:48) ) p (cid:35) where [ · ] is the integer part.Proof. By definition of the L ( T ) norm, we have (cid:88) n i (cid:54) = n j (cid:12)(cid:12) ˆ U (cid:48) ( n i − n j ) (cid:12)(cid:12) / ≤ R (Λ) (cid:88) r ∈ D ( V ) (cid:12)(cid:12) ˆ U (cid:48) ( r ) (cid:12)(cid:12) / = R (Λ) sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) U (cid:48) h ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) : || h || L ( T ) ≤ , Spec( h ) ⊂ D ( V ) (cid:27) (B.2)Now, since D ( V ) is a Λ( p (cid:48) )-system, we can bound the right hand side of (B.2) usingH¨older’s inequality as follows:RHS(B.2) ≤ µ ( U (cid:48) ) − p (cid:48) sup (cid:110) || h || L p (cid:48) ( T ) : || h || L ( T ) ≤ , Spec( h ) ⊂ D ( V ) (cid:111) ≤ ˜ Cµ ( U (cid:48) ) − p (cid:48) , for some constant ˜ C = ˜ C ( V, p (cid:48) ) > µ ( U (cid:48) ) (cid:88) n i (cid:54) = n j (cid:12)(cid:12) ˆ U (cid:48) ( n i − n j ) (cid:12)(cid:12) ≤ ˜ C R (Λ) µ ( U (cid:48) ) − p , as required. (cid:3) Claim B.4.
Let g ∈ L ( T ) with Spec( g ) ⊂ V = { n i } i , and U ⊂ T be a measurable subset.If ν ( U ) ≥ R ( V ) / (4 R ( V ) + 1) then || g || L ( T ) ≤ µ ( U ) (cid:90) U | g ( x ) | dx. Proof.
First, we can write g = (cid:88) i ˆ g ( n i ) z n i , so that, separating the diagonal terms from the others, we have (cid:90) U | g ( x ) | dx = µ ( U ) (cid:88) i | ˆ g ( n i ) | + (cid:88) i (cid:54) = j ˆ U ( n i − n j )ˆ g ( n i )ˆ g ( n j )= µ ( U ) || g || L ( T ) + (cid:104) Q U g, g (cid:105) , (B.3)where Q U = ( q ij ) is an operator on L ( T ) with matrix representation, in the base { z i } ,given by q ij = (cid:40) ˆ U ( n i − n j ) n i (cid:54) = n j . (B.4) ODAL LENGTH OF TORAL EIGENFUNCTIONS 22
Since U ( · ) is real-valued, ˆ U ( − n ) = ˆ U , thus Q U is a self-adjoint operator whose Hilbert-Schmidt norm is bounded by || Q U || ≤ R ( V ) / (cid:32)(cid:88) n (cid:54) =0 (cid:12)(cid:12) ˆ U ( n ) (cid:12)(cid:12) (cid:33) / = ( R ( V ) µ ( U )(1 − µ ( U ))) / . (B.5)In particular, if µ ( U ) ≥ R ( V ) / (4 R ( V ) + 1), we have ( R ( V ) µ ( U )(1 − µ ( U ))) / ≤ µ ( U ) / (cid:3) We are finally ready to prove Theorem B.1:
Proof of Theorem B.1.
Let 0 < ν ≤ R ( V ) / (4 R ( V ) + 1) be some parameter and denoteby A ( ν ) the smallest constant such that || g || L ( T ) ≤ A ( ν ) (cid:90) U | g ( x ) | dx, for all g ∈ L ( T ) with Spec( g ) ⊂ V and any set U ⊂ T with µ ( U ) ≥ ν . Moreover, let ϕ ( ν ) = log A ( ν ), ∆( ν ) = ν p (cid:48) (4 B ) − and m be given by Claim (B.3). Applying LemmaB.2, bearing in mind that m ≤ Bν − p , we obtain a set U ⊂ T of measure µ ( U ) ≥ ν +∆( ν )such that (cid:90) U | g ( x ) | dx ≤ (cid:18) CBν p (cid:48) (cid:19) Bν − p (cid:90) U | g ( x ) | dx. Since, by definition of A ( · ), || g || L ( T ) ≤ A ( ν + ∆( ν )) (cid:90) U | g ( x ) | dx, we have A ( ν ) ≤ A ( ν + ∆( ν )) (cid:18) CBν p (cid:48) (cid:19) Bν − p (cid:48) , and taking the logarithm of both sides, we finally deduce ϕ ( ν ) − ϕ ( ν + ∆( ν ))∆( ν ) ≤ B ν p (cid:48) log CB ν p (cid:48) ≤ C ( ε, B ) ν p (cid:48) + ε . (B.6)Comparing (B.6) with the differential inequality dϕ ( ν ) /dν ≤ C ( ε ) Bν − − p (cid:48) − ε and bearingin mind (B.3), we deduce that ϕ ( ν ) ≤ C ( ε ) ˜ C R (Λ) ν − − p (cid:48) − ε , as required. If ν ( U ) ≥ R ( V ) / (4 R ( V ) + 1), then Claim B.4 shows that the conclusion ofTheorem B.1 is still satisfied. (cid:3) B.3.
Proof of Lemma B.2.
In this section we prove Lemma B.2. The proof followsclosely the arguments in [27, Section 3.4], again we claim no originality. We will need thefollowing definition:
ODAL LENGTH OF TORAL EIGENFUNCTIONS 23
Definition B.5.
Let m be a positive integer and let τ, κ > g ∈ L ( T ), we say that g ∈ EP n loc ( τ, κ ) if for every t ∈ (0 , τ ) there exist constants a ( t ) , ..., a m ( t ) ∈ C such that (cid:80) k | a k | = 1 and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =0 a k ( t ) f kt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( T ) ≤ κ , where f kt ( · ) := f ( e ( kt ) · ) . We refer the reader to [27, Section 3.1-3.4] for an accurate description of the class EP m loc ( τ, κ ). Intuitively, functions in EP n loc ( τ, κ ) “behave like”trigonometric polynomialsof degree m . The key estimate that we will need is the following [27, Corollary 3.5’]: Lemma B.6.
Let g ∈ EP m loc ( τ, κ ) for some integer m > and some τ, κ > . Moreover,let U ⊂ T be a set of positive measure and ν := µ ( e ( nτ ) U \ U ) . There exists a set U ⊃ U of measure µ ( U \ U ) ≥ ν such that (cid:90) U | g ( x ) | dx ≤ (cid:18) Cm ν (cid:19) m (cid:18)(cid:90) U | g ( x ) | dx + κ (cid:19) . We will also need the following two claims:
Claim B.7.
Let U ⊂ T be a measurable subset and let m be as in Lemma B.2. Then,there exists a subspace V m of L ( T ) of dimension at most m such that for all g ∈ L ( T ) orthogonal to V m , we have || g || L ( T ) ≤ µ ( U (cid:48) ) (cid:90) U (cid:48) | g ( x ) | dx for all subsets U (cid:48) ⊂ U with µ ( U (cid:48) ) ≥ µ ( U ) / Proof.
Indeed, let | σ | ≤ | σ | ≤ ... be the eigenvalues of the operator Q U (cid:48) defined in (B.4)with U (cid:48) instead of U . Then we take V m to be the subspace generated by the eigenvectorswith eigenvalues σ , ..., σ m . We are now going to show that V m has the claimed property.By definition of m , we have (cid:88) i | σ i | = || Q U (cid:48) || ≤ (cid:88) i (cid:54) = j | q ij | = (cid:88) n i (cid:54) = n j (cid:12)(cid:12) ˆ U (cid:48) ( n i − n j ) (cid:12)(cid:12) ≤ µ ( U (cid:48) ) ( m + 1)4 . Thus, | σ m +1 | ≤ m + 1 · µ ( U (cid:48) ) ( m + 1)4 ≤ µ ( U (cid:48) ) . Therefore Claim B.7 follows from that fact that the norm of Q U (cid:48) restricted to L ( T ) \ V m is at most | σ m +1 | ≤ µ ( U (cid:48) ) / (cid:3) Now, if V is finite we let N = | V | , if V is infinite we can ignore the dependence on N in the rest of the argument. With this notation, we claim the following: Claim B.8.
Let U ⊂ T be a measurable set, m be as in Lemma B.2 and, if V is finite,suppose that m < N , moreover let g ∈ L ( T ) with Spec( g ) ⊂ V . Then there exists some σ ∈ [0 , such that g ∈ EP m loc ( τ, κ ) where κ = µ ( U ) ( m + 1) (cid:82) U | g ( x ) | dx , τ = σ/ m and,moreover ν := µ ( e ( mτ ) U \ U ) ≥ µ ( U ) / m . ODAL LENGTH OF TORAL EIGENFUNCTIONS 24
Proof.
Let t ∈ [0 ,
1) be given, since exponentials with different frequencies are linearlyindependent in L ( T ), we can choose coefficients a k ( t ), so that (cid:80) k | a k | = 1 and thefunction h ( · ) = m (cid:88) k =0 a k ( t ) g kt ( · ) , where g kt ( x ) = (cid:80) a n i e ( nkt ) e ( n i x ), is orthogonal to V m , given in Claim B.7, provided that m < N . Therefore, Claim B.7 gives || h || L ( T ) ≤ µ ( U (cid:48) ) (cid:90) U (cid:48) | h ( x ) | dx, (B.7)for all U (cid:48) ⊂ U with µ ( U (cid:48) ) ≥ µ ( U ) / U (cid:48) in order to estimate the RHS of(B.7). Let t ≥ U (cid:48) = U t := ∩ mk =0 e ( − kt ) U , since the function t → µ ( U t \ U ) iscontinuous and takes value 0 at t = 0, we can find some sufficiently small τ > t ∈ (0 , τ ), the set U t := ∩ mk =0 e ( − kt ) U has measure at least µ ( U ) /
2. To estimatethe RHS of (B.7), we observe that, for every k = 0 , ..., m , we have (cid:90) U t | g kt ( x ) | dx ≤ (cid:90) e ( − kt ) U | g kt ( x ) | dx = (cid:90) U | g ( x ) | dx. Thus, the Cauchy-Schwartz inequality gives (cid:90) U t | h ( x ) | dx ≤ (cid:32) m (cid:88) k =0 (cid:90) U t | g kt ( x ) | dx (cid:33) ≤ ( m + 1) (cid:90) U | g ( x ) | dx. (B.8)Hence, (B.7) together with (B.8), bearing in mind that µ ( U t ) ≥ µ ( U ) /
2, give that for all t ∈ (0 , τ ) there exists coefficients a ( t ) , ..., a m ( t ) such that (cid:80) k | a k | = 1 and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =0 a k ( t ) g kt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( T ) ≤ m + 1) µ ( U ) (cid:90) U | g ( x ) | dx. We are now left with proving the claimed estimates on τ and ν . Let ψ ( s ) = µ ( e ( s ) U \ U ),bearing in mind that µ ( U ) ≤ R ( V ) / (4 R ( V ) + 1) so that, by (B.2), µ ( T \ U ) ≥ (4 R ( V ) +1) − ≥ (2 m ) − , we have (cid:90) ψ ( s ) ds = µ ( U ) µ ( T \ U ) ≥ µ ( U )2 m . Thus, since ψ ( s ) is non-negative and continuous, there exists some σ ∈ [0 ,
1] such that forall s ≤ σ we have µ ( e ( s ) U \ U ) ≤ µ ( U ) / m . We now verify that such τ = σ/m satisfies µ ( U t ) ≥ µ ( U ) / t ∈ (0 , τ ). Indeed, bearing in mind that kt ∈ (0 , mτ ), we have µ ( U t ) = µ ( ∩ mk =0 e ( − kt ) U ) ≥ µ ( U ) − m (cid:88) k =1 µ ( e ( kt ) U \ U ) ≥ µ ( U ) − m µ ( U )2 m ≥ µ ( U ) / , (B.9)concluding the proof of Clam B.8. (cid:3) We are finally ready to present the proof of Lemma B.2 Suppose that n i (cid:54) = n j for i (cid:54) = j and (cid:80) i a n i e ( n i x ) = 0. Multiplying both sides by e ( − n x ) andintegrating for x ∈ T , we see that a = 0. Repeating the argument, we get a i = 0 for all i . ODAL LENGTH OF TORAL EIGENFUNCTIONS 25
Proof of Lemma B.2.
Suppose that m < N , then, applying Lemma B.6 with the choice ofparameters given by Claim B.8, we obtain part (1) of Lemma B.2. For part (2), LemmaB.6 gives (cid:90) U | g ( x ) | dx ≤ (cid:18) Cm µ ( U ) (cid:19) m (cid:18) m + 1) µ ( U ) + 1 (cid:19) (cid:90) U | g ( x ) | dx ≤ (cid:18) Cm µ ( U ) (cid:19) m (cid:90) U | g ( x ) | dx, (B.10)as required.Let us now suppose that m ≥ N , then the Nazarov-Tur´an Lemma [27, Theorem 1], forany set U ⊂ T of measure µ ( U ) = µ ( U ) + µ ( U ) / m , gives (cid:90) U | g ( x ) | dx ≤ (cid:18) Cµ ( U ) µ ( U ) (cid:19) N − (cid:90) U | g ( x ) | dx ≤ (cid:18) C + C m (cid:19) N − (cid:90) U | g ( x ) | dx and (B.10) follows. (cid:3) References [1]
J.-M. Aza¨ıs and M. Wschebor , Level sets and extrema of random processes and fields , JohnWiley & Sons, Inc., Hoboken, NJ, 2009.[2]
M. V. Berry , Regular and irregular semiclassical wavefunctions , Journal of Physics A: Mathemat-ical and General, 10 (1977), p. 2083.[3] ,
Semiclassical mechanics of regular and irregular motion , Les Houches lecture series, 36 (1983),pp. 171–271.[4]
M. V. Berry , Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections,fluctuations, curvature , J. Phys. A, 35 (2002), pp. 3025–3038.[5]
P. Billingsley , Convergence of probability measures , John Wiley & Sons, 2013.[6]
E. Bombieri and J. Bourgain , A problem on sums of two squares , Int. Math. Res. Not. IMRN,(2015), pp. 3343–3407.[7]
J. Bourgain , On toral eigenfunctions and the random wave model , Israel Journal of Mathematics,201 (2014), pp. 611–630.[8]
J. Buckley and I. Wigman , On the number of nodal domains of toral eigenfunctions , in AnnalesHenri Poincar´e, vol. 17, Springer, 2016, pp. 3027–3062.[9]
V. Cammarota, O. Klurman, and I. Wigman , Boundary effect on the nodal length for arithmeticrandom waves, and spectral semi-correlations , Comm. Math. Phys., 376 (2020), pp. 1261–1310.[10]
S. Y. Cheng , Eigenfunctions and nodal sets , Comment. Math. Helv., 51 (1976), pp. 43–55.[11]
H. Donnelly and C. Fefferman , Nodal sets of eigenfunctions on reimannian manifolds , Inven-tiones mathematicae, 93 (1988), pp. 161–183.[12]
P. Erd˝os and R. R. Hall , On the angular distribution of Gaussian integers with fixed norm ,vol. 200, 1999, pp. 87–94. Paul Erd˝os memorial collection.[13]
L. C. Evans , Partial differential equations , vol. 19 of Graduate Studies in Mathematics, AmericanMathematical Society, Providence, RI, 1998.[14]
A. Granville and I. Wigman , Planck-scale mass equidistribution of toral Laplace eigenfunctions ,Comm. Math. Phys., 355 (2017), pp. 767–802.[15]
P. Humphries , Equidistribution in shrinking sets and L -norm bounds for automorphic forms ,Math. Ann., 371 (2018), pp. 1497–1543.[16] I. K´atai and I. K¨ornyei , On the distribution of lattice points on circles , Ann. Univ. Sci. Budapest.E¨otv¨os Sect. Math., 19 (1976), pp. 87–91 (1977).[17]
M. Krishnapur, P. Kurlberg, and I. Wigman , Nodal length fluctuations for arithmetic randomwaves , Ann. of Math. (2), 177 (2013), pp. 699–737.
ODAL LENGTH OF TORAL EIGENFUNCTIONS 26 [18]
P. Kurlberg and I. Wigman , On probability measures arising from lattice points on circles , Math.Ann., 367 (2017), pp. 1057–1098.[19]
S. Lester and Z. Rudnick , Small scale equidistribution of eigenfunctions on the torus , Comm.Math. Phys., 350 (2017), pp. 279–300.[20]
A. Logunov , Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorffmeasure , Ann. of Math. (2), 187 (2018), pp. 221–239.[21] ,
Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower boundin Yau’s conjecture , Ann. of Math. (2), 187 (2018), pp. 241–262.[22]
A. Logunov and E. Malinnikova , Nodal sets of Laplace eigenfunctions: estimates of the Haus-dorff measure in dimensions two and three , in 50 years with Hardy spaces, vol. 261 of Oper. TheoryAdv. Appl., Birkh¨auser/Springer, Cham, 2018, pp. 333–344.[23]
A. Logunov and E. Malinnikova , Lecture notes on quantitative unique continuation for solutionsof second order elliptic equations , arXiv: Analysis of PDEs, (2019).[24]
R. ´Alvaro and S. Andrea , Nodal set of monochromatic waves satisfying the random wave model ,Arxiv preprint: https://arxiv.org/abs/2011.03467, (2020).[25]
D. Mangoubi , Local asymmetry and the inner radius of nodal domains , Comm. Partial DifferentialEquations, 33 (2008), pp. 1611–1621.[26]
F. Nazarov and M. Sodin , Asymptotic laws for the spatial distribution and the number of con-nected components of zero sets of gaussian random functions , J. Math. Phys. Anal. Geom., 12 (2016),pp. 205–278.[27]
F. L. Nazarov , Local estimates for exponential polynomials and their applications to inequalitiesof the uncertainty principle type , Algebra i Analiz, 5 (1993), pp. 3–66.[28] ,
Summability of large powers of logarithm of classic lacunary series and its simplest con-sequences , Unpublished, Preprint available at https://users.math.msu.edu/users/fedja/prepr.html,(1995).[29]
F. Oravecz, Z. Rudnick, and I. Wigman , The Leray measure of nodal sets for random eigen-functions on the torus , Ann. Inst. Fourier (Grenoble), 58 (2008), pp. 299–335.[30]
A. Sartori , On the fractal structure of attainable probability measures , Bull. Pol. Acad. Sci. Math.,66 (2018), pp. 123–133.[31] ,
Planck-scale number of nodal domains for toral eigenfunctions , J. Funct. Anal., 279 (2020),pp. 108663, 22.[32]
I. Wigman and N. Yesha , Central limit theorem for Planck-scale mass distribution of toral Laplaceeigenfunctions , Mathematika, 65 (2019), pp. 643–676.[33]
S.-T. Yau , Open problems in geometry , J. Ramanujan Math. Soc., 15 (2000), pp. 125–134.[34]
S. Zelditch , Eigenfunctions of the Laplacian on a Riemannian manifold , vol. 125 of CBMS RegionalConference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences,Washington, DC; by the American Mathematical Society, Providence, RI, 2017.
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