Communications in Mathematical Physics

, Volume 358, Issue 3, pp 895–917 | Cite as

Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

  • Naser T. Sardari


Let Q(X) be any integral primitive positive definite quadratic form in k variables, where \({k\geq4}\), and discriminant D. For any integer n, we give an upper bound on the number of integral solutions of Q(X) = n in terms of n, k, and D. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus \({\mathbb{T}^d}\) for \({d\geq 5}\). This conjecture is motivated by the work of Berry [2,3] on the semiclassical eigenfunction hypothesis.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.University of Wisconsin-MadisonMadisonUSA

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