On a problem of geometry of numbers arising in spectral theory

TL;DR

This paper investigates lattice point counting in anisotropically expanding domains and derives asymptotic formulas with remainder estimates, applying results to eigenvalue distributions of Laplace operators on flat tori.

Contribution

It introduces new asymptotic formulas for lattice points in anisotropic domains and applies these to spectral theory of Laplace operators on flat tori.

Findings

Derived leading asymptotic term for lattice point counts.

Established remainder estimates under various conditions.

Applied results to eigenvalue distribution in spectral theory.

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.