Lp Norms of the Lattice Point Discrepancy

Abstract

We estimate the Lp norms of the discrepancy between the volume and the number of integer points in rΩ - x, a dilated by a factor r and translated by a vector x of a convex body Ω in Rd with smooth boundary with strictly positive curvature, {∫R∫Td|∑k∈ZdχrΩ-x(k)-rd|Ω||pdxdμ(r-R)}1/p,where μ is a Borel measure compactly supported on the positive real axis and R→ + ∞.