Fourier analytic techniques for lattice point discrepancy

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 is locally flat of finite order. We consider both integer points problems and irregularities of distribution problems. The paper is self-contained. The results on irregularities of distribution are new.