Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

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Let Q(X) be any integral primitive positive definite quadratic form in k variables, where k≥ 4 , 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 Td for d≥ 5. This conjecture is motivated by the work of Berry [2,3] on the semiclassical eigenfunction hypothesis.

Original languageEnglish (US)
Pages (from-to)895-917
Number of pages23
JournalCommunications In Mathematical Physics
Issue number3
StatePublished - Mar 1 2018

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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