Distribution of similar configurations in subsets of ${\mathbb{F}}_q^d$

Abstract

Let ${\mathbb{F}}_q$ be a finite field of order $q$ and $E$ be a set in ${\mathbb{F}}_q^d$. The distance set of $E$ is defined by $\Delta (E):=\{\Vert x-y\Vert :x,y\in E\}$, where $\Vert \alpha \Vert ={\alpha }_1^2+\cdots +{\alpha }_d^2$. Iosevich, Koh and Parshall (2018) proved that if $d\ge 2$ is even and $|E|\ge 9q^{d/2}$, then $${\mathbb{F}}_q=\frac{\Delta (E)}{\Delta (E)}=\left\{\frac{a}{b}:a\in \Delta (E),b\in \Delta (E)\setminus \{0\}\right\}.$$ In other words, for each $r\in {\mathbb{F}}_q^{\ast }$ there exist $(x,y)\in E^2$ and $(x^{\prime },y^{\prime })\in E^2$ such that $\Vert x-y\Vert \ne 0$ and $\Vert x^{\prime }-y^{\prime }\Vert =r\Vert x-y\Vert$. Geometrically, this means that if the size of $E$ is large, then for any given $r\in {\mathbb{F}}_q^{\ast }$ we can find a pair of edges in the complete graph $K_{|E|}$ with vertex set $E$ such that one of them is dilated by $r\in {\mathbb{F}}_q^{\ast }$ with respect to the other. A natural question arises whether it is possible to generalize this result to arbitrary subgraphs of $K_{|E|}$ with vertex set $E$ and this is the goal of this paper. In this paper, we solve this problem for $k$-paths $(k\ge 2)$, simplexes and 4-cycles. We are using a mix of tools from different areas such as enumerative combinatorics, group actions and Turán type theorems.

26 pages, 6 figures. Corrected typos. Theorems 1.7 and 1.8 have been improved. Published in Discrete Mathematics