The number of integer points in a family of anisotropically expanding domains

TL;DR

This paper studies the count of integer points in anisotropically expanding domains, providing new estimates for the remainder term and applying these results to improve eigenvalue distribution estimates on foliated tori.

Contribution

It introduces new remainder estimates for integer point counts in anisotropically expanding domains and applies these to enhance eigenvalue distribution bounds on foliated tori.

Findings

Established bounds for the remainder in integer point counting.

Improved eigenvalue distribution estimates in the adiabatic limit.

Analyzed average remainder over rotations in SO(n).

Abstract

We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and expand in the directions, orthogonal to this subspace. We prove some estimates for the remainder, imposing additional assumptions on the boundary of the domain. We study the average remainder estimates, where the averages are taken over rotated images of the domain by a subgroup of the group SO(n) of orthogonal transformations of the Euclidean space R^n. Using these results, we improve the remainder estimate in the adiabatic limit formula for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in the particular case when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.