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

Abstract

In this paper, we study problems about the similar configurations in ${\mathbb{F}}_q^d$. Let $G=(V,E)$ be a graph, where $V=\{1,2,\dots ,n\}$ and $E\subseteq \left(\genfrac{}{}{0ex}{}{V}{2}\right)$. 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 ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_n\in \mathcal{E}$ and distinct ${\mathbf{y}}_1,{\mathbf{y}}_2,\dots ,{\mathbf{y}}_n\in \mathcal{E}$ such that $\Vert {\mathbf{y}}_i-{\mathbf{y}}_j\Vert =r\Vert {\mathbf{x}}_i-{\mathbf{x}}_j\Vert \ne 0$ whenever $\{i,j\}\in E$, where $\Vert \mathbf{x}\Vert :=x_1^2+x_2^2+\cdots +x_d^2$ for $\mathbf{x}=(x_1,x_2,\dots ,x_d)\in {\mathbb{F}}_q^d$. We show that if $\mathcal{E}$ has size at least $C_k{q}^{d/2}$, 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.

19 pages; any comments are welcome