An improved $L^2$ restriction theorem in finite fields

TL;DR

This paper generalizes the finite field restriction theorem by replacing uniform Fourier bounds with $L^p$ bounds, leading to improved results and applications to Sidon sets and Hamming varieties.

Contribution

It introduces a new restriction estimate in finite fields using $L^p$ bounds, surpassing previous uniform bound results by Mockenhaupt and Tao.

Findings

Improved restriction range in many cases

Applications to Sidon sets

Applications to Hamming varieties

Abstract

Mockenhaupt and Tao (Duke 2004) proved a finite field analogue of the Stein--Tomas restriction theorem, establishing a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure $\mu$ on a vector space over a finite field. Their result is expressed in terms of exponents that describe uniform bounds on the measure and its Fourier transform. We generalise this result by replacing the uniform bounds on the Fourier transform with suitable $L^p$ bounds, and we show that our result improves upon the Mockenhaupt--Tao range in many cases. We also provide a number of applications of our result, including to Sidon sets and Hamming varieties.