An asymptotic formula is proved for the number of integral points in a family of bounded domains with smooth boundary in Euclidean space; these domains remain unchanged along some linear subspace and expand in the directions orthogonal to this subspace. A sharper estimate for the remainder is obtained in the case where the domains are strictly convex. These results make it possible to improve the remainder estimate in the adiabatic limit formula (due to the first author) for the eigenvalue

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... ibution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in the particular case where the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus. §1. Statement of the problem and the main results The classical problem on integer point distributions is to study the asymptotic behavior of the number of points of the integer lattice Z n in a family of homothetic domains in R n . This problem originated with the Gauss problem on the number of integer points in a disk (directly related to the arithmetic problem on the number of representations of an integer as a sum of two squares) and is studied sufficiently well (see, e.g., the books [3, 2, 4, 8] and the references therein). In this paper we investigate the much less studied problem on counting the integer points in a family of anisotropically expanding domains. More precisely, let F be a pdimensional linear subspace of R n and H = F ⊥ the q-dimensional orthogonal complement of F with respect to the standard inner product (·, ·) in R n , p + q = n. For any ε > 0, consider the following linear transformation T ε : R n → R n : For any bounded domain S in R n with smooth boundary, we put Our main goal in this paper is to study the asymptotic behavior of the function n ε (S) as ε → 0. Before stating the main results of the paper, we introduce some auxiliary notions. Let Γ = Z n ∩ F ; Γ is a free Abelian group. We denote by r = rank Γ ≤ p the rank of Γ. For r ≥ 1, denote by ( 1 , 2 , . . . , r ) some basis in Γ. Let V be the r-dimensional subspace of R n spanned by the vectors ( 1 , 2 , . . . , r ) . Observe that Γ is a lattice in V . Denote by Q ⊂ V the parallelepiped spanned by the vectors ( 1 , 2 , . . . , r ) and by |Q| 2010 Mathematics Subject Classification. Primary 11P21; Secondary 58J50.