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

Steven Senger, Dung The Tran

TL;DR

The paper introduces a sharp finite-field point-sphere incidence bound for (4,s)-Salem sets, showing that a pseudorandom configuration with s in (1/4,1/2] and size |P| \ll q^{d/(4s)} yields near-random incidence behavior: |I(P,S) - |P||S|/q| \ll q^{d/4} |P|^{1-s} |S|^{3/4}. The core technique lifts P to a higher-dimensional set P' and converts spheres to hyperplanes, applying a nontrivial incidence bound for (4,s)-Salem sets, with a separate treatment for zero- versus nonzero-constant-term cases, ensuring sharpness through explicit constructions. The authors additionally extend the method to even moments (u,s)-Salem sets and discuss implications for unit distances, sum-product phenomena, and connections to Sidon-type sets and cone-extension frameworks, highlighting both theoretical interest and potential applications in additive combinatorics and finite-field geometry.

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, dot-product configurations, and sum-product type phenomena, and we extend the method to \((u,s)\)-Salem sets for even moments \(u\ge4\).

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

TL;DR

The paper introduces a sharp finite-field point-sphere incidence bound for (4,s)-Salem sets, showing that a pseudorandom configuration with s in (1/4,1/2] and size |P| \ll q^{d/(4s)} yields near-random incidence behavior: |I(P,S) - |P||S|/q| \ll q^{d/4} |P|^{1-s} |S|^{3/4}. The core technique lifts P to a higher-dimensional set P' and converts spheres to hyperplanes, applying a nontrivial incidence bound for (4,s)-Salem sets, with a separate treatment for zero- versus nonzero-constant-term cases, ensuring sharpness through explicit constructions. The authors additionally extend the method to even moments (u,s)-Salem sets and discuss implications for unit distances, sum-product phenomena, and connections to Sidon-type sets and cone-extension frameworks, highlighting both theoretical interest and potential applications in additive combinatorics and finite-field geometry.

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 is a \((4,s)\)-Salem set with and , then for any finite family of spheres in , 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, dot-product configurations, and sum-product type phenomena, and we extend the method to \((u,s)\)-Salem sets for even moments .
Paper Structure (10 sections, 21 theorems, 177 equations)

This paper contains 10 sections, 21 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}
  3. Sharpness example
  4. Sharpness examples of the incidence theorem
  5. An explicit $(4,\frac{1}{2})$-Salem set
  6. Extensions with $(u,s)$-Salem sets
  7. Further applications and discussion
  8. A relaxed notion of Sidon sets
  9. A sum--product application via point--sphere incidences for $(4,s)$-Salem sets
  10. On the Koh-Lee-Pham framework via cone extension estimates

Key Result

Lemma 1.3

Let $s\in[0,1]$ and $k\ge1$. The $(2k,s)$-Salem condition is equivalent to the additive energy bound

Theorems & Definitions (45)

  • Definition 1.1: Fourier transform and norms
  • Definition 1.2
  • Lemma 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6: Koh-Lee-Pham, KLP22
  • Theorem 1.7
  • Remark 1.1
  • Corollary 1.8: Unit distances in $(4,s)$-Salem sets
  • proof
  • ...and 35 more