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 dilated by a factor $r$ and translated by a vector $x$ of a convex body $\Omega$ in ${\mathbb{R}}^d$, ${\left\{{\int }_{{\mathbb{T}}^d}{\left(\frac{1}{H}{\int }_R^{R+H}{\left|{\sum }_{k\in {\mathbb{Z}}^d}{\chi }_{r\Omega -x}(k)-r^d\left|\Omega \right|\right|}^{2}dr\right)}^{p/2}dx\right\}}^{1/p}.$ We obtain estimates for fixed values of $H$ and $R\to \infty$, and also asymptotic estimates when $H\to \infty$.

References in corpus (2)

• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic
• $L^p$ and Weak-$L^p$ estimates for the number of integer points in translated domains

Cited by in corpus (2)

• $L^p$ norms of the lattice point discrepancy
• Higher Moments for Lattice Point Discrepancy of Convex Domains and Annuli