$L^{p}$ and Weak-$L^{p}$ estimates for the number of integer points in translated domains

TL;DR

This paper extends recent results by estimating the $L^{p}$ and Weak-$L^{p}$ norms of the discrepancy between volume and integer points count in translated domains, providing new bounds in harmonic analysis.

Contribution

It introduces new bounds for discrepancy estimates in $L^{p}$ and Weak-$L^{p}$ spaces, extending previous work by M. Huxley on integer point counting.

Findings

Derived bounds for discrepancy norms in $L^{p}$ spaces.

Extended Huxley's results to Weak-$L^{p}$ norms.

Provided sharper estimates for translated domains.

Abstract

Revisiting and extending a recent result of M.Huxley, we estimate the $L^{p}\left( \mathbb{T}^{d}\right) $ and Weak-$L^{p}\left( \mathbb{T}^{d}\right) $ norms of the discrepancy between the volume and the number of integer points in translated domains.