  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.
