$L^{p}$ Norms of the Lattice Point Discrepancy

Abstract

We estimate the $L^{p}$ norms of the discrepancy between the volume and the number of integer points in $r\Omega -x$, a dilated by a factor r and translated by a vector x of a convex body $\Omega$ in ${\mathbb {R}}^{d}$ with smooth boundary with strictly positive curvature,

$$\begin{aligned} \left\{ {\displaystyle \int _{{\mathbb {R}}}}{\displaystyle \int _{{\mathbb {T}}^{d}}}\left| \sum _{k\in {\mathbb {Z}}^{d}}\chi _{r\Omega -x}(k)-r^{d}\left| \Omega \right| \right| ^{p}dxd\mu (r-R) \right\} ^{1/p}, \end{aligned}$$

where $\mu$ is a Borel measure compactly supported on the positive real axis and $R\rightarrow +\infty$.