$L^{p}$ and Weak-$L^{p}$ estimates for the number of integer points in translated domains

Abstract

Revisiting and extending a recent result of M.Huxley, we estimate the $L^{p}\left( \mathbb{T}^{d}\right) $ and Weak-$L^{p}\left( \mathbb{T}^{d}\right) $ norms of the discrepancy between the volume and the number of integer points in translated domains.