Elementary methods for incidence problems in finite fields

TL;DR

This paper employs elementary techniques to establish an incidence theorem for points and spheres in finite fields, demonstrating that large point sets determine many circles, akin to Beck's Theorem, with optimal bounds up to constants.

Contribution

It introduces elementary methods to prove incidence results in finite fields and extends Beck's Theorem to circles with near-optimal bounds.

Findings

Proved an incidence theorem for points and spheres in finite fields.

Large point sets determine a positive proportion of all circles.

Results are optimal up to multiplicative constants.

Abstract

We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb{F}_q^n$. As an application, we show that any point set of $P\subset \mathbb{F}_q^2$ with $|P|\geq 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck's Theorem for circles which is optimal up to multiplicative constants.