An improved L2 restriction theorem in finite fields

Jonathan M. Fraser; Firdavs Rakhmonov

doi:10.1090/proc/17707

An improved L² restriction theorem in finite fields

Research output: Contribution to journal › Article › peer-review

Abstract

Mockenhaupt and Tao [Duke Math. J. 121 (2004), no. 1, 35–74] proved a finite field analogue of the Stein–Tomas restriction theorem, establishing a range of q for which L^q → L²restriction 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^pbounds, 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.

Original language	English
Number of pages	14
Journal	Proceedings of the American Mathematical Society
Volume	Ahead of print
Early online date	12 Jun 2026
DOIs	https://doi.org/10.1090/proc/17707
Publication status	E-pub ahead of print - 12 Jun 2026

Keywords

Restriction problem
Fourier transform
Finite fields

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10.1090/proc/17707

Fraser_2026_PAMS_Improved-L2-restriction-finite-fields_AAM_CCBY
Copyright © 2026 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1090/proc/17707
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A refined fourier analysis: A refined fourier analysis
Fraser, J. (PI)
Leverhulme Trust
1/09/24 → 29/02/28
Project: Standard
Fourier analytic tachniques in finitie: Fourier analytic techniques in finite fields
Fraser, J. (PI)
EPSRC
1/05/24 → 30/04/25
Project: Standard

Cite this

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title = "An improved L2 restriction theorem in finite fields",

abstract = "Mockenhaupt and Tao [Duke Math. J. 121 (2004), no. 1, 35–74] proved a finite field analogue of the Stein–Tomas restriction theorem, establishing a range of q for which Lq → L2 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 Lp 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.",

keywords = "Restriction problem, Fourier transform, Finite fields",

author = "Fraser, \{Jonathan M.\} and Firdavs Rakhmonov",

note = "Funding: The first author was financially supported by a Leverhulme Trust Research Project Grant (RPG-2023-281), an EPSRC Standard Grant (EP/Y029550/1), and an EPSRC Open Fellowship (EP/Z533440/1). The second author was financially supported by a Leverhulme Trust Research Project Grant (RPG-2023-281).",

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AU - Fraser, Jonathan M.

AU - Rakhmonov, Firdavs

N1 - Funding: The first author was financially supported by a Leverhulme Trust Research Project Grant (RPG-2023-281), an EPSRC Standard Grant (EP/Y029550/1), and an EPSRC Open Fellowship (EP/Z533440/1). The second author was financially supported by a Leverhulme Trust Research Project Grant (RPG-2023-281).

PY - 2026/6/12

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N2 - Mockenhaupt and Tao [Duke Math. J. 121 (2004), no. 1, 35–74] proved a finite field analogue of the Stein–Tomas restriction theorem, establishing a range of q for which Lq → L2 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 Lp 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.

AB - Mockenhaupt and Tao [Duke Math. J. 121 (2004), no. 1, 35–74] proved a finite field analogue of the Stein–Tomas restriction theorem, establishing a range of q for which Lq → L2 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 Lp 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.

KW - Restriction problem

KW - Fourier transform

KW - Finite fields

U2 - 10.1090/proc/17707

DO - 10.1090/proc/17707

M3 - Article

SN - 0002-9939

VL - Ahead of print

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

ER -

An improved L2 restriction theorem in finite fields

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Keywords

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A refined fourier analysis: A refined fourier analysis

Fourier analytic tachniques in finitie: Fourier analytic techniques in finite fields

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An improved L² restriction theorem in finite fields