Mixed $L^{p}(L^{2})$ norms of the lattice point discrepancy

Abstract:

We estimate some mixed $L^{p}\left ( L^{2}\right )$ norms of the discrepancy between the volume and the number of integer points in $r\Omega -x$, a dilation by a factor $r$ and a translation by a vector $x$ of a convex body $\Omega$ in $\mathbb {R}^{d}$ with smooth boundary with nonvanishing Gaussian curvature, \[ \left \{ {\displaystyle \int _{\mathbb {T}^{d}}}\left ( \dfrac {1}{H}{\displaystyle \int _{R}^{R+H}}\left \vert \sum _{k\in \mathbb {Z}^{d}}\chi _{r\Omega -x}(k)-r^{d}\left \vert \Omega \right \vert \right \vert ^{2}dr\right ) ^{p/2}dx\right \} ^{1/p}. \] We obtain estimates for fixed values of $H$ and $R\to \infty$, and also asymptotic estimates when $H\to \infty$.