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

TL;DR

This paper establishes a new bound on incidences between points and varieties over finite fields using spectral methods, leading to improved results in finite geometry and distance problems.

Contribution

It introduces a novel incidence bound over finite fields by analyzing the incidence matrix via group algebra techniques, advancing the understanding of geometric incidences.

Findings

New incidence bound for points and flats in finite geometries

Improved results for pinned distance problems under weaker conditions

Characterization of singular values and vectors of incidence matrices

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.