Integer points in domains and adiabatic limits

TL;DR

This paper develops an asymptotic formula for counting integer points in certain stretched domains and applies it to improve eigenvalue distribution estimates in adiabatic limits on foliated manifolds.

Contribution

It introduces a new asymptotic counting method for integer points in domains with specific geometric properties and refines eigenvalue distribution estimates in adiabatic limits.

Findings

Asymptotic formula for integer points in stretched domains with smooth boundaries.

Enhanced remainder estimates for strictly convex domains.

Improved eigenvalue distribution bounds in adiabatic limits for torus foliations.

Abstract

We prove an asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space with smooth boundary, which remain unchanged along some linear subspace and stretch out in the directions, orthogonal to this subspace. A more precise estimate for the remainder is obtained in the case when the domains are strictly convex. Using these results, we improved the remainder estimate in the adiabatic limit formula (due to the first author) 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 a particular case when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.