Geometric structures in pseudo-random graphs

TL;DR

This paper develops a framework for counting geometric structures in pseudo-random graphs, bridging discrete geometry, measure theory, and graph theory, with applications to finite field analogs of continuous problems.

Contribution

It introduces a general method for analyzing geometric configurations in pseudo-random graphs, improving existing results in finite field geometric problems.

Findings

Recovered and improved several finite field geometric results

Established new connections between discrete geometry and graph theory

Provided a versatile framework applicable to various geometric counting problems

Abstract

In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.