Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 1, 2021

On the variance of the nodal volume of arithmetic random waves

  • Giacomo Cherubini EMAIL logo and Niko Laaksonen
From the journal Forum Mathematicum


Rudnick and Wigman (2008) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the d-dimensional torus is O(E/𝒩), as E, where E is the energy and 𝒩 is the dimension of the eigenspace corresponding to E. Previous results have established this with stronger asymptotics when d=2 and d=3. In this brief note we prove an upper bound of the form O(E/𝒩1+α(d)-ϵ), for any ϵ>0 and d4, where α(d) is positive and tends to zero with d. The power saving is the best possible with the current method (up to ϵ) when d5 due to the proof of the 2-decoupling conjecture by Bourgain and Demeter.

Communicated by Valentin Blomer

Funding statement: This work was supported by ERC Consolidator Grant 648017 and by the Rényi Intézet Lendület Automorphic Research Group.


We thank Miklós Abért for making us interested in this problem, and Gergely Harcos for useful discussions. We also thank Riccardo Maffucci for his comments on an earlier version of the paper. We thank the anonymous referee for the careful reading of the paper.


[1] J.-M. Azaïs and M. Wschebor, Level Sets and Extrema of Random Processes and Fields, John Wiley & Sons, Hoboken, 2009. 10.1002/9780470434642Search in Google Scholar

[2] J. Benatar and R. W. Maffucci, Random waves on 𝕋3: Nodal area variance and lattice point correlations, Int. Math. Res. Not. IMRN 2019 (2019), no. 10, 3032–3075. 10.1093/imrn/rnx220Search in Google Scholar

[3] M. V. Berry, Statistics of nodal lines and points in chaotic quantum billiards: Perimeter corrections, fluctuations, curvature, J. Phys. A 35 (2002), no. 13, 3025–3038. 10.1088/0305-4470/35/13/301Search in Google Scholar

[4] E. Bombieri and J. Bourgain, A problem on sums of two squares, Int. Math. Res. Not. IMRN 2015 (2015), no. 11, 3343–3407. 10.1093/imrn/rnu005Search in Google Scholar

[5] J. Bourgain, On toral eigenfunctions and the random wave model, Israel J. Math. 201 (2014), no. 2, 611–630. 10.1007/s11856-014-1037-zSearch in Google Scholar

[6] J. Bourgain and C. Demeter, New bounds for the discrete Fourier restriction to the sphere in 4D and 5D, Int. Math. Res. Not. IMRN 2015 (2015), no. 11, 3150–3184. 10.1093/imrn/rnu036Search in Google Scholar

[7] J. Bourgain and C. Demeter, The proof of the l2 decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351–389. 10.4007/annals.2015.182.1.9Search in Google Scholar

[8] V. Cammarota, Nodal area distribution for arithmetic random waves, Trans. Amer. Math. Soc. 372 (2019), no. 5, 3539–3564. 10.1090/tran/7779Search in Google Scholar

[9] P. Erdős and R. R. Hall, On the angular distribution of Gaussian integers with fixed norm, Discrete Math. 200 (1999), 87–94. 10.1016/S0012-365X(98)00329-XSearch in Google Scholar

[10] O. M. Fomenko, Uniform distribution of lattice points on multidimensional ellipsoids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 154 (1986), no. 7, 144–153, 179. Search in Google Scholar

[11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Elsevier/Academic, Amsterdam, 2007. Search in Google Scholar

[12] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 1 (1918), no. 4, 357–376. 10.1007/BF01465095Search in Google Scholar

[13] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), no. 1–2, 11–51. 10.1007/BF01202991Search in Google Scholar

[14] H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, American Mathematical Society, Providence, 1997. 10.1090/gsm/017Search in Google Scholar

[15] I. Kátai and I. Környei, On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 19 (1976), 87–91. Search in Google Scholar

[16] M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves, Ann. of Math. (2) 177 (2013), no. 2, 699–737. 10.4007/annals.2013.177.2.8Search in Google Scholar

[17] Y. V. Linnik, Ergodic Properties of Algebraic Fields, Ergeb. Math. Grenzgeb. (3) 45, Springer, New York Inc, 1968. 10.1007/978-3-642-86631-9Search in Google Scholar

[18] A. V. Malyšev, On the representation of integers by positive quadratic forms, Trudy Mat. Inst. Steklov 65 (1962), 1–212. Search in Google Scholar

[19] D. Marinucci, G. Peccati, M. Rossi and I. Wigman, Non-universality of nodal length distribution for arithmetic random waves, Geom. Funct. Anal. 26 (2016), no. 3, 926–960. 10.1007/s00039-016-0376-5Search in Google Scholar

[20] F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 299–335. 10.5802/aif.2351Search in Google Scholar

[21] A. Palczewski, J. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal. 34 (1997), no. 5, 1865–1883. 10.1137/S0036142995289007Search in Google Scholar

[22] C. Pommerenke, Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden, Acta Arith. 5 (1959), 227–257. 10.4064/aa-5-2-227-257Search in Google Scholar

[23] C. Pommerenke, Berichtigung zu meiner Arbeit “Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden”, Acta Arith. 7 (1961/62), 279. Search in Google Scholar

[24] Z. Rudnick and I. Wigman, On the volume of nodal sets for eigenfunctions of the Laplacian on the torus, Ann. Henri Poincaré 9 (2008), no. 1, 109–130. 10.1007/s00023-007-0352-6Search in Google Scholar

Received: 2021-11-16
Revised: 2021-05-20
Published Online: 2021-12-01
Published in Print: 2022-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 31.5.2023 from
Scroll to top button