Extension Theorems for Spheres in the Finite Field Setting

TL;DR

This paper investigates the boundedness of extension operators for spheres over finite fields, improving Tomas-Stein exponents by analyzing incidences and Fourier transforms in even dimensions.

Contribution

It introduces new bounds for extension operators on finite field spheres and refines previous results using analytic and Fourier analysis techniques.

Findings

Improved Tomas-Stein exponents for finite field spheres

Established bounds on incidences between points and spheres

Derived explicit Fourier transform formulas for spheres

Abstract

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, our previous results. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.