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

TL;DR

This paper extends previous results on the distribution of distances in large subsets of finite fields to more complex subgraphs, including paths, simplexes, and cycles, using combinatorial and algebraic tools.

Contribution

It generalizes known distance distribution results to arbitrary subgraphs like paths, simplexes, and cycles in finite field vector spaces.

Findings

Established distribution properties for k-paths, simplexes, and 4-cycles.

Used combinatorics, group actions, and Turán theorems in proofs.

Extended geometric distance results to complex subgraph configurations.

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 $\Delta(E):=\{\lVert x-y \rVert :x,y\in E\}$, where $\lVert \alpha \rVert=\alpha_1^2+\dots+\alpha_d^2$. Iosevich, Koh and Parshall (2018) proved that if $d\geq 2$ is even and $|E|\geq 9q^{d/2}$, 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 $r\in \mathbb{F}_q^*$ there exist $(x,y)\in E^2$ and $(x',y')\in E^2$ such that $\lVert x-y\rVert\neq0$ and $\lVert x'-y' \rVert=r\lVert x-y\rVert$. Geometrically, this means that if the size of $E$ is large, then for any given $r \in \mathbb{F}_q^*$ 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 $r\in \mathbb{F}_q^*$ 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 $(k\geq 2)$, simplexes and 4-cycles. We are using a mix of tools from different areas such as enumerative combinatorics, group actions and Tur\'an type theorems.