Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields

TL;DR

This paper improves bounds for embedding certain graphs in subsets of finite field vector spaces, challenging the previous reliance on maximum degree as a complexity measure and proposing refined exponents.

Contribution

It introduces new results that surpass previous bounds by refining the complexity measure for graph embeddings in finite field vector spaces.

Findings

Improved embedding bounds with better exponents

Evidence that maximum degree is not always the best complexity measure

Progress towards identifying the correct complexity notion for graph embeddings

Abstract

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C q^{m+\frac{d-1}{2}}$, for any $t>0$, there exist $k+1$ points $x^1, \dots, x^{k+1}$ in $E$ such that $||x^i-x^j||=t$ if the $i$'th vertex is connected to the $j$'th vertex by an edge in $G$. In this paper, we give several indications that the maximum degree is not always the right notion of complexity and prove several concrete results to obtain better exponents than the Iosevich-Parshall result affords. This can be viewed as a step towards understanding the right notion of complexity for graph embeddings in subsets of vector spaces over finite fields.