$L^p$ 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, \[ \left\{ {\displaystyle\int_{\mathbb R}}{\displaystyle\int_{\mathbb{T}^{d}}}\left\vert \sum_{k\in\mathbb{Z}^{d}}χ_{rΩ-x}(k)-r^{d}\left\vert Ω\right\vert \right\vert ^{p}dxdμ(r-R) \right\} ^{1/p}, \] where $\mu$ is a Borel measure compactly supported on the positive real axis and $R\to +\infty$.

37 pages, 6 figures. arXiv admin note: text overlap with arXiv:1706.04419

References in corpus (2)

• $L^p$ and Weak-$L^p$ estimates for the number of integer points in translated domains
• Mixed $L^p(L^2)$ norms of the lattice point discrepancy

Cited by in corpus (1)

• Higher Moments for Lattice Point Discrepancy of Convex Domains and Annuli