  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.
