Fourier analytic techniques for lattice point discrepancy

TL;DR

This paper uses Fourier analysis to study the discrepancy in counting integer points within certain convex bodies that have flat points, providing elementary proofs for these specific geometric cases.

Contribution

It offers a detailed analysis of lattice point discrepancy for convex bodies with flat boundary points, focusing on a specific class where the boundary resembles a power function.

Findings

Provides explicit discrepancy estimates for convex bodies with flat points

Develops elementary proofs for the lattice point problem in these cases

Enhances understanding of distribution irregularities in geometric number theory

Abstract

Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes (or convex polytopes). In this paper we provide a detailed description of several discrepancy problems in the particular planar case where the boundary coincides locally with the graph of the function $\mathbb{R\ni}t\mapsto\left\vert t\right\vert ^{\gamma}$, with $\gamma>2$. We consider both \textit{integer points} problems and \textit{irregularities of distribution} problems. The above \textquotedblleft restriction\textquotedblright\ to a particular family of convex bodies is compensated by the fact that many proofs are elementary. The paper is entirely self-contained.