On the limiting behaviour of arithmetic toral eigenfunctions
Annales de l'Institut Fourier, Online first, 59 p.

We consider a wide class of families (F m ) m of Gaussian fields on 𝕋 d = d / d defined by

F m :x1 |Λ m | λΛ m ξ λ e i2πλ,x

where the ξ λ ’s are independent standard normals and Λ m is the set of solutions λ d to the equation p(λ)=m for some fixed elliptic polynomial p with integer coefficients. The case p(x)=x 1 2 ++x d 2 amounts to considering a random Laplace eigenfunction whose law is sometimes called the arithmetic random wave and has been studied in the past by many authors. In contrast, we consider three classes of polynomials p: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except the multiples of x 1 2 +x 2 2 +x 3 2 , and a wide family of polynomials in many variables.

For these three classes of polynomials, we study the (d-1)-dimensional volume 𝒱 m of the zero set of F m . We compute the asymptotics, as m+ along certain well chosen subsequences of integers, of the expectation and variance of 𝒱 m . Moreover, we prove that in the same limit, 𝒱 m -𝔼[𝒱 m ] Var(𝒱 m ) converges to a standard normal.

As in previous analogous works on this topic for the arithmetic random wave, a very general method reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to p(λ)=m. More precisely, we need to study the number of such solutions for a fixed m, as well as the number of quadruples of solutions (λ,μ,ν,ι) satisfying λ+μ+ν+ι=0, a.k.a. 4-correlations, and the rate of convergence of the (rescaled) counting measure of Λ m towards a certain limiting measure on the hypersurface {p(x)=1}. To this end, we use many previous results on this topic but also prove a new estimate on correlations which may be of independent interest.

On considère une large classe de familles (F m ) m de champs gaussiens sur 𝕋 d = d / d définies par

F m :x1 |Λ m | λΛ m ξ λ e i2πλ,x

où les ξ λ des variables normales standard indépendantes et Λ m est l’ensemble des solutions λ d de l’équation p(λ)=m pour un certain polynôme fixé p à coefficients entiers. Le cas p(x)=x 1 2 ++x d 2 revient à considérer une fonction propre aléatoire du laplacien dont la loi est parfois appelée onde arithmétique aléatoire et a été étudiée par de nombreux auteurs. À l’inverse, on considère trois classes de polynômes p : une famille de formes quadratiques définies positives en deux variables, toutes les formes quadratiques définies positives en trois variables sauf les multiples de x 1 2 +x 2 2 +x 3 2 , et une large famille de polynômes en plusieurs variables.

Pour ces trois classes de polynômes, on étudie le volume 𝒱 m en dimension d-1 de l’ensemble des zéros de F m . On calcule les asymptotiques, quand m+, le long de certaines sous-suites d’entiers bien choisies, de l’espérance et de la variance de 𝒱 m . De plus, on montre que dans la même limite, 𝒱 m -𝔼[𝒱 m ] Var(𝒱 m ) converge vers une loi normale standard.

Comme dans des travaux antérieurs analogues sur ce sujet pour l’onde arithmétique aléatoire, une méthode très générale réduit le problème de ces asymptotiques à l’étude de certaines propriétés arithmétiques de l’ensemble des solutions de p(λ)=m. Plus précisément, on étudie le nombre de telles solutions pour m fixé, ainsi que le nombre de quadruplets de solutions (λ,μ,ν,ι) vérifiant λ+μ+ν+ι=0, également appelées 4-corrélations, et le taux de convergence de la mesure de comptage (mise à l’échelle) de Λ m vers une mesure limite sur l’hypersurface {p(x)=1}. Pour cela, on utilise de nombreux résultats antérieurs sur le sujet mais on établit aussi une nouvelle estimation sur les corrélations qui présente son propre intérêt.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3630
Classification: 11D72, 28C20, 35P20, 60G60, 11D45, 11P21
Keywords: Gaussian fields, Limiting theorems, Lattice points on manifolds, Equidistribution, Lattice point correlations, Kac–Rice formulas, Wiener chaos.
Mot clés : Champs gaussiens, théorèmes limite, réseaux de points sur les variétés, équidistribution, corrélations de points de réseaux, formules de Kac–Rice, chaos de Wiener.

Maffucci, Riccardo 1; Rivera, Alejandro 1

1 EPFL, MA SB Batiment 8, Lausanne (Switzerland)
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Maffucci, Riccardo; Rivera, Alejandro. On the limiting behaviour of arithmetic toral eigenfunctions. Annales de l'Institut Fourier, Online first, 59 p.

[1] Azaïs, Jean-Marc; Wschebor, Mario Level sets and extrema of random processes and fields, John Wiley & Sons, 2009, xii+393 pages | DOI | MR | Zbl

[2] Beliaev, Dmitry; Cammarota, Valentina; Wigman, Igor Two point function for critical points of a random plane wave, Int. Math. Res. Not. (2019) no. 9, pp. 2661-2689 | DOI | MR | Zbl

[3] Benatar, Jacques; Maffucci, Riccardo W. Random waves on 𝕋 3 : nodal area variance and lattice point correlations, Int. Math. Res. Not. (2019) no. 10, pp. 3032-3075 | DOI | MR | Zbl

[4] Bérard, Pierre Volume des ensembles nodaux des fonctions propres du laplacien, Sémin. Théor. Spectr. Géom., Volume 3 (1984-1985), 4, 9 pages | MR | Zbl

[5] Berry, Michael V. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J. Phys. A. Math. Gen., Volume 35 (2002) no. 13, pp. 3025-3038 | DOI | MR | Zbl

[6] Birch, Bryan J. Forms in many variables, Proc. R. Soc. Lond., Ser. A, Volume 265 (1961), pp. 245-263 | DOI | MR | Zbl

[7] Bombieri, Enrico; Bourgain, Jean A problem on sums of two squares, Int. Math. Res. Not. (2015) no. 11, pp. 3343-3407 | DOI | MR | Zbl

[8] Bombieri, Enrico; Pila, Jonathan The number of integral points on arcs and ovals, Duke Math. J., Volume 59 (1989) no. 2, pp. 337-357 | DOI | MR | Zbl

[9] Bourgain, Jean; Demeter, Ciprian New bounds for the discrete Fourier restriction to the sphere in four and five dimensions (2013) (https://arxiv.org/abs/1310.5244)

[10] Bourgain, Jean; Demeter, Ciprian The proof of the l 2 decoupling conjecture, Ann. Math., Volume 182 (2015) no. 1, pp. 351-389 | DOI | MR | Zbl

[11] Brüning, Jochen; Gromes, Dieter Über die Länge der Knotenlinien schwingender Membranen, Math. Z., Volume 124 (1972), pp. 79-82 | DOI | MR | Zbl

[12] Cammarota, Valentina Nodal area distribution for arithmetic random waves, Trans. Am. Math. Soc., Volume 372 (2019) no. 5, pp. 3539-3564 | DOI | MR | Zbl

[13] Cammarota, Valentina; Marinucci, Domenico; Wigman, Igor On the distribution of the critical values of random spherical harmonics, J. Geom. Anal., Volume 26 (2016) no. 4, pp. 3252-3324 | DOI | MR | Zbl

[14] Canzani, Yaiza; Hanin, Boris Local universality for zeros and critical points of monochromatic random waves (2016) (https://arxiv.org/abs/1610.09438)

[15] Canzani, Yaiza; Toth, John A. Nodal sets of Schrödinger eigenfunctions in forbidden regions, Ann. Henri Poincaré, Volume 17 (2016) no. 11, pp. 3063-3087 | DOI | MR | Zbl

[16] Castryck, Wouter; Cluckers, Raf; Dittmann, Philip; Nguyen, Kien Huu The dimension growth conjecture, polynomial in the degree and without logarithmic factors (2019) (https://arxiv.org/abs/1904.13109)

[17] Cheng, Shiu Yuen Eigenfunctions and nodal sets, Comment. Math. Helv., Volume 51 (1976) no. 1, pp. 43-55 | DOI | MR | Zbl

[18] Cherubini, Giacomo; Laaksonen, Niko On the variance of the nodal volume of arithmetic random waves (2020) (https://arxiv.org/abs/2007.12143)

[19] Cilleruelo, Javier; Córdoba, Antonio Trigonometric polynomials and lattice points, Proc. Am. Math. Soc., Volume 115 (1992) no. 4, pp. 899-905 | DOI | MR | Zbl

[20] Cilleruelo, Javier; Córdoba, Antonio Lattice points on ellipses, Duke Math. J., Volume 76 (1994) no. 3, pp. 741-750 | DOI | MR | Zbl

[21] Dalmao, Federico; Estrade, Anne; León, José On 3-dimensional Berry’s model (2019) (https://arxiv.org/abs/1912.09774)

[22] Dias, Dimitri The angular distribution of integral ideal numbers with a fixed norm in quadratic extensions (2014) (https://arxiv.org/abs/1404.6271)

[23] Donnelly, Harold; Fefferman, Charles Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., Volume 93 (1988) no. 1, pp. 161-183 | DOI | MR | Zbl

[24] Duke, William; Rudnick, Zeév; Sarnak, Peter Density of integer points on affine homogeneous varieties, Duke Math. J., Volume 71 (1993) no. 1, pp. 143-179 | DOI | MR | Zbl

[25] El-Hajj, Layan; Toth, John A. Intersection bounds for nodal sets of planar Neumann eigenfunctions with interior analytic curves, J. Differ. Geom., Volume 100 (2015) no. 1, pp. 1-53 | DOI | MR | Zbl

[26] Erdős, Pál; Hall, Richard R. On the angular distribution of Gaussian integers with fixed norm, Discrete Math., Volume 200 (1999) no. 1-3, pp. 87-94 (Pál Erdős memorial collection) | DOI | MR | Zbl

[27] Fomenko, Oleg M. On the uniform distribution of integral points on multidimensional ellipsoids, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Volume 154 (1986), pp. 144-153 | Zbl

[28] Hartshorne, Robin Algebraic geometry. Corr. 3rd printing, Graduate Texts in Mathematics, 52, Springer, 1983 | Zbl

[29] Heath-Brown, D. Roger The density of rational points on curves and surfaces, Ann. Math., Volume 155 (2002) no. 2, pp. 553-595 | DOI | MR | Zbl

[30] Heath-Brown, D. Roger Counting rational points on algebraic varieties, Analytic number theory (Lecture Notes in Mathematics), Volume 1891, Springer, 2006, pp. 51-95 | DOI | MR | Zbl

[31] Hörmander, Lars The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Mathematics, Springer, 2003, x+440 pages | DOI | MR | Zbl

[32] Iwaniec, Henryk Fourier coefficients of modular forms of half-integral weight, Invent. Math., Volume 87 (1987) no. 2, pp. 385-401 | DOI | MR | Zbl

[33] Iwaniec, Henryk Topics in classical automorphic forms, Graduate Studies in Mathematics, 17, American Mathematical Society, 1997, xii+259 pages | DOI | MR | Zbl

[34] Krishnapur, Manjunath; Kurlberg, Pär; Wigman, Igor Nodal length fluctuations for arithmetic random waves, Ann. Math., Volume 177 (2013) no. 2, pp. 699-737 | DOI | MR | Zbl

[35] Maffucci, Riccardo W. Nodal intersections for random waves against a segment on the 3-dimensional torus, J. Funct. Anal., Volume 272 (2017) no. 12, pp. 5218-5254 | DOI | MR | Zbl

[36] Magyar, Ákos Diophantine equations and ergodic theorems, Am. J. Math., Volume 124 (2002) no. 5, pp. 921-953 | DOI | MR | Zbl

[37] Magyar, Ákos On the distribution of solutions to Diophantine equations, A panorama of discrepancy theory (Lecture Notes in Mathematics), Volume 2107, Springer, 2014, pp. 487-538 | DOI | MR | Zbl

[38] Mahler, Kurt Note on Hypothesis K of Hardy and Littlewood, J. Lond. Math. Soc., Volume 11 (1936) no. 2, pp. 136-138 | DOI | MR | Zbl

[39] Marinucci, Domenico; Peccati, Giovanni; Rossi, Maurizia; Wigman, Igor Non-universality of nodal length distribution for arithmetic random waves, Geom. Funct. Anal., Volume 26 (2016) no. 3, pp. 926-960 | DOI | MR | Zbl

[40] Mudgal, Akshat Additive energies on spheres (2021) (https://arxiv.org/abs/2105.06925)

[41] Nourdin, Ivan; Peccati, Giovanni Normal approximations with Malliavin calculus. From Stein’s method to universality, Cambridge Tracts in Mathematics, 192, Cambridge University Press, 2012, xiv+239 pages | DOI | MR

[42] Oravecz, Ferenc; Rudnick, Zeév; Wigman, Igor The Leray measure of nodal sets for random eigenfunctions on the torus, Ann. Inst. Fourier, Volume 58 (2008) no. 1, pp. 299-335 | DOI | Numdam | MR | Zbl

[43] Pila, Jonathan Density of integral and rational points on varieties, Columbia University Number Theory Seminar (New York, 1992) (Astérisque), Société Mathématique de France, 1995 no. 228, pp. 183-187 | Numdam | MR | Zbl

[44] Rudnick, Zeév; Wigman, Igor On the volume of nodal sets for eigenfunctions of the Laplacian on the torus, Ann. Henri Poincaré, Volume 9 (2008) no. 1, pp. 109-130 | DOI | MR | Zbl

[45] Rudnick, Zeév; Wigman, Igor; Yesha, Nadav Nodal intersections for random waves on the 3-dimensional torus, Ann. Inst. Fourier, Volume 66 (2016) no. 6, pp. 2455-2484 | DOI | Numdam | MR | Zbl

[46] Sarnak, Peter Some applications of modular forms, Cambridge Tracts in Mathematics, 99, Cambridge University Press, 1990, x+111 pages | DOI | MR | Zbl

[47] Toth, John A.; Zelditch, Steve Counting nodal lines which touch the boundary of an analytic domain, J. Differ. Geom., Volume 81 (2009) no. 3, pp. 649-686 | DOI | MR | Zbl

[48] Wigman, Igor Fluctuations of the nodal length of random spherical harmonics, Commun. Math. Phys., Volume 298 (2010) no. 3, pp. 787-831 | DOI | MR | Zbl

[49] Yau, Shing-Tung Survey on partial differential equations in differential geometry, Seminar on Differential Geometry (Annals of Mathematics Studies), Volume 102, Princeton University Press, 1982, pp. 3-71 | MR | Zbl

[50] Yau, Shing-Tung Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) (Proceedings of Symposia in Pure Mathematics), Volume 54, American Mathematical Society (1993), pp. 1-28 | DOI | MR | Zbl

[51] Zelditch, Steve Eigenfunctions and nodal sets, Geometry and topology. Lectures given at the geometry and topology conferences at Harvard University, Cambridge, MA, USA, April 29–May 1, 2011 and Lehigh University, Bethlehem, PA, USA, May 25–27, 2012 (Surveys in Differential Geometry), Volume 18, International Press, 2013, pp. 237-308 | DOI | MR | Zbl

[52] Zygmund, Antoni On Fourier coefficients and transforms of functions of two variables, Stud. Math., Volume 50 (1974), pp. 189-201 | DOI | MR | Zbl

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