Fourier transforms of indicator functions, lattice point discrepancy, and the stability of integrals

TL;DR

This paper establishes sharp Fourier transform estimates for indicator functions of sets with real analytic boundaries, explores lattice point discrepancies, and investigates the stability of integrals under perturbations, connecting harmonic analysis and number theory.

Contribution

It provides new sharp estimates for Fourier transforms of indicator functions and lattice point discrepancies, and introduces a stability theorem for the growth rate of real analytic functions near zeros.

Findings

Sharp Fourier transform estimates for sets with real analytic boundary

Nontrivial lattice point discrepancy results

Stability of growth rates of real analytic functions near zeros

Abstract

We prove sharp estimates for Fourier transforms of indicator functions of bounded open sets in ${\mathbb R}^n$ with real analytic boundary, as well as nontrivial lattice point discrepancy results. Both will be derived from estimates on Fourier transforms of hypersurface measures. Relations with maximal averages are discussed, connecting two conjectures of Iosevich and Sawyer from [ISa1]. We also prove a theorem concerning the stability under function perturbations of the growth rate of a real analytic function near a zero. This result is sharp in an appropriate sense. It implies a corresponding stability result for the local integrablity of negative powers of a real analytic function near a zero.