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

Jonathan M. Fraser, Firdavs Rakhmonov

TL;DR

The paper advances finite-field restriction theory by replacing uniform Fourier bounds with $L^p$-averaged bounds on the Fourier transform of the surface measure. It proves an explicit $L^q$ extension bound for $f o f\, ext{restricted}$ in terms of $eta_p$ and $ abla ext{F}$-dimension parameters, recovering Mockenhaupt–Tao at $p= fty$ and often improving the range for finite $p$. The approach leverages an $L^{(2p)'} o L^{2p}$ estimate via $ orm{p}{ar}$ and interpolation, with applications to Salem-set geometry, Sidon sets, and Hamming varieties. The results yield concrete, geometry-dependent restrictions—e.g., products of spheres, sphere-zeros, and cutoff cylinders—along with meaningful consequences for Sidon sets and Hamming varieties in finite field settings, broadening the toolkit for discrete restriction theory and additive combinatorics.

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 $μ$ 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.

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

TL;DR

The paper advances finite-field restriction theory by replacing uniform Fourier bounds with -averaged bounds on the Fourier transform of the surface measure. It proves an explicit extension bound for in terms of and -dimension parameters, recovering Mockenhaupt–Tao at and often improving the range for finite . The approach leverages an estimate via and interpolation, with applications to Salem-set geometry, Sidon sets, and Hamming varieties. The results yield concrete, geometry-dependent restrictions—e.g., products of spheres, sphere-zeros, and cutoff cylinders—along with meaningful consequences for Sidon sets and Hamming varieties in finite field settings, broadening the toolkit for discrete restriction theory and additive combinatorics.

Abstract

Mockenhaupt and Tao (Duke 2004) proved a finite field analogue of the Stein--Tomas restriction theorem, establishing a range of for which restriction estimates hold for a given measure 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 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.
Paper Structure (12 sections, 11 theorems, 78 equations)

This paper contains 12 sections, 11 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. The restriction problem and overview of results
  3. Discrete Fourier analysis and the restriction problem over finite fields
  4. Main results
  5. Proof of Theorem \ref{['main']}
  6. Simple examples
  7. Products of spheres
  8. The sphere of radius zero
  9. Cutoff cylinders
  10. Applications
  11. Large Sidon sets
  12. Hamming varieties

Key Result

Theorem 1.1

Let $\mu$ be a nonzero, finite, compactly supported Borel measure on $\mathbb{R}^d$, and let $0<\alpha,\beta<d$. Suppose that for all $x\in \mathbb{R}^d$ and all $r>0$, and for all $\xi\in \mathbb{R}^d$, Then holds for all functions $f\in L^2(\mu)$ and all $q\geqslant 2+\frac{4(d-\alpha)}{\beta}$.

Theorems & Definitions (15)

  • Theorem 1.1: Stein--Tomas
  • Theorem 1.2: Mockenhaupt--Tao
  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['converse']}
  • Proposition 4.1
  • Corollary 4.2
  • Corollary 5.1
  • proof : Proof of Corollary \ref{['sidonrestriction']}
  • ...and 5 more