A sharp point-sphere incidence bound for $(u, s)$-Salem sets

TL;DR

This paper proves a new sharp point-sphere incidence bound over finite fields for -salem sets, improving classical bounds and applying additive energy techniques and lifting arguments.

Contribution

It introduces a novel incidence bound for -salem sets in finite fields, extending the framework to -salem sets for even moments and refining bounds for related geometric problems.

Findings

Established a sharp incidence bound for -salem sets in finite fields.

Improved classical point-sphere incidence bounds across a broad parameter range.

Extended the method to -salem sets for even moments  and applications to unit distances and sum-product phenomena.

Abstract

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \((4,s)\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if \(P\subset \mathbb{F}_q^d\) is a \((4,s)\)-Salem set with \(s\in \big( \frac{1}{4}, \frac{1}{2} \big]\) and \(|P|\ll q^{ \frac{d}{4s}}\), then for any finite family \(S\) of spheres in \(\mathbb{F}_q^d\), \[ \bigg| I(P,S)-\frac{|P||S| }{q} \bigg| \ll q^{\frac{d}{4}}\,|P|^{1-s}\,|S|^{\frac{3}{4}}. \] This estimate improves the classical point-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the \((4,s)\)-Salem property. As applications, we derive refined bounds for unit distances and sum-product type phenomena, and we extend the method to \((u,s)\)-Salem sets for even moments \(u\ge4\).