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

## TL;DR

This paper investigates mixed $L^p(L^2)$ norms of the discrepancy between the volume and lattice points in dilated and translated convex bodies, providing estimates for large parameters and asymptotic behavior.

## Contribution

It introduces new bounds for mixed norms of lattice point discrepancy, extending understanding of their asymptotic properties in high-dimensional convex geometry.

## Key findings

- Derived estimates for fixed $H$ as $R o \infty$
- Obtained asymptotic estimates when $H o \\infty$
- Enhanced understanding of discrepancy behavior in convex bodies

## 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\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 $H\to\infty$.