Lattice points in large bodies

Let B be a body in s-dimensional Euclidean space with smooth boundary throughout, and let t be a large real parameter by which the body B is dilated ("blown up"). The number of integer points in the enlarged body tB is approximated in first order by the volume of tB. During many decades, the question for the error involved in this approximation has created a deep and prolific mathematical theory. While in the planar case (s=2) the situation is comparatively well understood, for bodies of our familiar three-dimensional space the matter remains much more enigmatic. Therefore, the main objective of the project submitted is to estimate, from above and below, this "lattice point discrepancy" (number of integer points minus volume) of general three-dimensional bodies tB, and to obtain also results about its "average" behaviour. Particular attention will be paid to three-dimensional bodies which are invariant with respect to rotation around one of the coodinate axes. Furthermore, the case is to be investigated in detail that the boundary of K contains curves on which the curvature of the surface vanishes identically. Similar investigations for special bodies of higher dimensions could lead to answers of questions concerning the arithmetic in certain generalizations of the complex field, like Hamilton`s quaternions.

Let B be a body in s-dimensional Euclidean space with smooth boundary throughout, and let t be a large real parameter by which the body B is dilated ("blown up"). The number of integer points in the enlarged body tB is approximated in first order by the volume of tB. During many decades, the question for the error involved in this approximation has created a deep and prolific mathematical theory. While in the planar case (s=2) the situation is comparatively well understood, for bodies of our familiar three-dimensional space the matter remains much more enigmatic. Therefore, the main objective of the project submitted is to estimate, from above and below, this "lattice point discrepancy" (number of integer points minus volume) of general three-dimensional bodies tB, and to obtain also results about its "average" behaviour. Particular attention will be paid to three-dimensional bodies which are invariant with respect to rotation around one of the coodinate axes. Furthermore, the case is to be investigated in detail that the boundary of K contains curves on which the curvature of the surface vanishes identically. Similar investigations for special bodies of higher dimensions could lead to answers of questions concerning the arithmetic in certain generalizations of the complex field, like Hamilton`s quaternions.