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.