Generalized point configurations in ${\mathbb F}_q^d$

TL;DR

This paper extends the study of distance graphs in finite field vector spaces by generalizing the notion of distance through bilinear and quadratic forms, establishing bounds for embedding various graph structures.

Contribution

It introduces a generalized framework for distances in finite fields and proves bounds for embedding complex graphs under this new setting.

Findings

Bounds on subset sizes for containing distance graphs

Embedding results for paths, trees, and cycles

Generalization of previous finite field distance graph results

Abstract

In this paper, we generalize \cite{IosevichParshall}, \cite{LongPaths} and \cite{cycles} by allowing the \emph{distance} between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of $\F_q^d$ for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.