The lattice discrepancy of large domains

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. The question for the error involved in this approximation - the so- called "lattice point discrepancy" - has created a deep and prolific mathematical theory. In the planar case (s = 2), the situation is comparatively well understood, as well as for bodies in three-dimensional space that are sort of "egg-shaped" (i.e., connvex with smooth boundary of nonzero curvature throghout, in mathematical terms). Therefore, the main objective of the project submitted is to focus on the most explicit deviation from convexity that a body can have, namely a "dent", the boundary being still smooth. It is expected that there will be obtained rather precise asymptotic formulas for the lattice discrepancy. These investigations are to be carried for in dimension 3 and bigger. The whole theme is a natural and central question of multivariate analysis. It can be viewed as sort of a basic science for multidimensional numerical integration, it has connections to Fourier theory, to the classic analytic theory of special arithmetic functions, to Diophantine equations, and to the theory of linear operators. Furthermore, there are even interactions, resp., applications to quantum physics and to crystallography.

The main issue of the project was to gain more precise insight about the asymptotic behavior of the number of number of integer points in large domains in the plane and in space, in particular about the difference ("discrepancy") between this number and the area, resp., volume of the domain under consideration. First of all, the tools and methods available for this type of questions have been substantially improved and refined. In the sequel, the results achieved have been applied to different special bodies. To mention just two examples, the ellipsoid of rotation and the three-dimensional torus ("doughnut") have been deal with under this aspect. Furthermore, it turned out that the methods developed permitted applications to problems in other mathematical disciplines, which at least in a far sense are based on an enumeration of points with integer coordinates. Last but not least, it may be mentioned that a new and fairly precise formula for the number of integer points in an arbitrary circular disk was used to carry out computer calculations which determined a certain mathematical constant (the so-called Masser-Gramain constant) with much higher precision than it was possible before.