A point-variety incidence theorem over finite fields, and its applications
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.
Cite
@article{arxiv.2408.10977,
title = {A point-variety incidence theorem over finite fields, and its applications},
author = {Xiangliang Kong and Itzhak Tamo},
journal= {arXiv preprint arXiv:2408.10977},
year = {2025}
}
Comments
23 pages. Compared to the earlier version, we delet a previously misquoted theorem (labeled as Theorem~1.6 in that version) and add a detailed summary of our proof strategy, along with several examples and remarks to further illustrate the incidence problem we address and our contributions