Some results on similar configurations in subsets of $\mathbb{F}_q^d$

TL;DR

This paper investigates the presence of similar geometric configurations in subsets of finite field vector spaces, establishing size thresholds for containing specific scaled structures like stars and paths.

Contribution

It introduces new bounds on subset sizes in finite fields that guarantee the existence of scaled geometric configurations, using combinatorial and graph-theoretic methods.

Findings

Subsets of size at least C_k q^{d/2} contain pairs of k-stars with a dilation ratio.

Subsets of size at least C·min{q^{(2d+1)/3}, max{q^3, q^{d/2}}} contain pairs of 4-paths with a dilation ratio.

The methods combine enumerative combinatorics and graph theory to derive these results.

Abstract

In this paper, we study problems about the similar configurations in $\mathbb{F}_q^d$. Let $G=(V, E)$ be a graph, where $V=\{1, 2, \ldots, n\}$ and $E\subseteq{V\choose2}$. For a set $\mathcal{E}$ in $\mathbb{F}_q^d$, we say that $\mathcal{E}$ contains a pair of $G$ with dilation ratio $r$ if there exist distinct $\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots, \boldsymbol{x}_n\in\mathcal{E}$ and distinct $\boldsymbol{y}_1, \boldsymbol{y}_2, \ldots, \boldsymbol{y}_n\in\mathcal{E}$ such that $\|\boldsymbol{y}_i-\boldsymbol{y}_{j}\|=r\|\boldsymbol{x}_i-\boldsymbol{x}_j\|\neq0$ whenever $\{i, j\}\in E$, where $\|\boldsymbol{x}\|:=x_1^2+x_2^2+\cdots+x_d^2$ for $\boldsymbol{x}=(x_1, x_2, \ldots, x_d)\in\mathbb{F}_q^d$. We show that if $\mathcal{E}$ has size at least $C_kq^{d/2}$, then $\mathcal{E}$ contains a pair of $k$-stars with dilation ratio $r$, and that if $\mathcal{E}$ has size at least $C\cdot\min\left\{q^{(2d+1)/3}, \max\left\{q^3, q^{d/2}\right\}\right\}$, then $\mathcal{E}$ contains a pair of $4$-paths with dilation ratio $r$. Our method is based on enumerative combinatorics and graph theory.