Fourier analytic techniques for lattice point discrepancy

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.