Integer points in domains and adiabatic limits

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.