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October 9, 2025Open Access

An improved L² restriction theorem in finite fields

Key Points

An improved restriction theorem enhances L^2 estimates in finite fields, applicable to various mathematical constructs.
Replacing uniform bounds with L^p bounds broadens the range of applicability and improves upon previous results.
The study employs advanced mathematical constructs from measure theory and Fourier analysis in finite fields.
Applications extend to mathematical structures like Sidon sets and Hamming varieties, emphasizing broader relevance.

Abstract

Mockenhaupt and Tao (Duke 2004) 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 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.

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