Lattice Points in Large Convex Bodies

Abstract.

 Denote by the number of points of the lattice in the “blown up” domain , where is a convex body in () whose boundary is smooth and has nonzero curvature throughout. It is proved that for every fixed

where for and . This improves a classic result of E. Hlawka [8] and its refinements due to E. Krätzel and W. G. Nowak ([14], [15]). The proof uses a multidimensional variant of the method of van der Corput for the estimation of exponential sums.