Incidences between points and generalized spheres over finite fields and related problems

Abstract

Let F q be a finite field of q elements where q is a large odd prime power andWe prove bounds on the number of incidences between a point set P and a Q-sphere set S, denoted by I(P, S), as the following.We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings Z q where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections 4 and 5, we prove a bound on the number of incidences between a random point set and a random Q-sphere set in F d q . We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

Elementary methods for incidence problems in finite fields

“…Let S be the set of all circles which contain less than or equal to 3 points from P ′ . We will show that (11) |S| < 5q 3 9 ,…”

“…The authors in [11] give an independent proof of Theorem 1 via graph theoretic methods similar to those used in [12]. Further applications of the incidence result are also given in [11].…”

“…It remains to prove (11), and to do this we will make use of the lower bound on I(P ′ , S) from Theorem 1. We have 5|S| − |P ′ | 1/2 |S| 1/2 q = |P ′ ||S| q − |P ′ | 1/2 |S| 1/2 q < I(P ′ , S) ≤ 2|S|, a rearrangement of which gives |S| < |P ′ |q 2 9 = 5q 3 9 , as required.…”

An improvement on the number of simplices in Fqd

“…is called a distinct distance subset if there are no four distinct points x, y, z, t ∈ U such that ||x − y|| = ||z − t||. In [10], Phuong et al studied the finite field analogue of this problem. More precisely, the authors of [10] proved that if |E| ≥ 2q (2d+1)/3 , then there exists a distinct distance subset of cardinality ≫ q 1/3 .…”

“…In [10], Phuong et al studied the finite field analogue of this problem. More precisely, the authors of [10] proved that if |E| ≥ 2q (2d+1)/3 , then there exists a distinct distance subset of cardinality ≫ q 1/3 . This implies that the result is only non-trivial when d ≥ 3.…”