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On a problem of geometry of numbers arising in spectral theory

Authors
  • Kordyukov, Yuri A.
  • Yakovlev, Andrey A.
Type
Preprint
Publication Date
Jul 22, 2015
Submission Date
Jul 22, 2015
Identifiers
DOI: 10.1134/S106192081504007X
Source
arXiv
License
Yellow
External links

Abstract

We study a lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains, which remain unchanged along some fixed linear subspace and expand in directions, orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.

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