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A point-variety incidence theorem over finite fields, and its applications

Xiangliang Kong, Itzhak Tamo

TL;DR

This work develops a spectral approach to incidence problems between points and a broad class of finite-field varieties by fully characterizing the singular-value decomposition of the point–variety incidence matrix through group-algebra methods. The authors derive sharp incidence bounds for point–variety pairs, with improvements in regimes where the variety family is relatively small and the graph is highly unbalanced. They apply the main bound to two geometric settings: incidences between points and n-flats, and pinned-distance problems, obtaining stronger results under certain parameter ranges and yielding corollaries that sharpen prior finite-field incidence bounds. The methodology combines SVD analysis, group-algebra eigenstructure, and Fourier-analytic techniques to obtain a unified framework for incidence problems with explicit dependence on field size, dimension, and the parametric form of varieties. This work thus advances understanding of how zero-eigenvalue structure influences incidence bounds and suggests avenues for further improvements in unbalanced geometric graphs.

Abstract

Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this paper, by characterizing the singular values and singular vectors of the corresponding incidence matrix through group algebras, we prove a bound on the number of incidences between points and varieties of a certain form over finite fields. Our result leads to a new incidence bound for points and flats in finite geometries, which improves previous results for certain parameter regimes. As another application of our point-variety incidence bound, we extend a result on pinned distance problems by Phuong, Thang, and Vinh, and independently by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev, under a weaker condition.

A point-variety incidence theorem over finite fields, and its applications

TL;DR

This work develops a spectral approach to incidence problems between points and a broad class of finite-field varieties by fully characterizing the singular-value decomposition of the point–variety incidence matrix through group-algebra methods. The authors derive sharp incidence bounds for point–variety pairs, with improvements in regimes where the variety family is relatively small and the graph is highly unbalanced. They apply the main bound to two geometric settings: incidences between points and n-flats, and pinned-distance problems, obtaining stronger results under certain parameter ranges and yielding corollaries that sharpen prior finite-field incidence bounds. The methodology combines SVD analysis, group-algebra eigenstructure, and Fourier-analytic techniques to obtain a unified framework for incidence problems with explicit dependence on field size, dimension, and the parametric form of varieties. This work thus advances understanding of how zero-eigenvalue structure influences incidence bounds and suggests avenues for further improvements in unbalanced geometric graphs.

Abstract

Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this paper, by characterizing the singular values and singular vectors of the corresponding incidence matrix through group algebras, we prove a bound on the number of incidences between points and varieties of a certain form over finite fields. Our result leads to a new incidence bound for points and flats in finite geometries, which improves previous results for certain parameter regimes. As another application of our point-variety incidence bound, we extend a result on pinned distance problems by Phuong, Thang, and Vinh, and independently by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev, under a weaker condition.
Paper Structure (17 sections, 25 theorems, 111 equations, 2 tables)

This paper contains 17 sections, 25 theorems, 111 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Problem settings and main results
  3. Application to point-flat incidences problem
  4. An application to the pinned distances problem
  5. Outline
  6. Preliminaries and notations
  7. The point-variety incidence graph
  8. Singular value decomposition
  9. Group algebra
  10. Proof of the main results
  11. Right singular vectors of the incidence matrix
  12. Left singular vectors of the incidence matrix
  13. Proof of Theorem \ref{['main_thm1']}
  14. Applications
  15. Point-flat incidences
  16. ...and 2 more sections

Key Result

Lemma 1.1

(Expander crossing lemma, Haemers95DSV12) Let $G$ be a bipartite graph with parts $A,B$ such that the vertices in $A$ all have degree $a$ and the vertices in $B$ all have degree $b$. Then, for any $X\subseteq A$ and $Y\subseteq B$, the number of edges between $X$ and $Y$, denoted by $e(X,Y)$, satisf where $\lambda_3$ is the third eigenvalue of $G$We denote $|\lambda_1| \geqslant |\lambda_2| \geqsl

Theorems & Definitions (47)

  • Lemma 1.1
  • Example 1
  • Example 2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • ...and 37 more