An inverse-type problem for cycles in local Cayley distance graphs

TL;DR

This paper investigates the existence of many cycles of a fixed even length in certain Cayley graphs over finite fields, providing an inverse result that guarantees a large number of such cycles under specific density conditions.

Contribution

It establishes an inverse-type theorem for cycles in local Cayley distance graphs, extending recent results by demonstrating abundant cycles through each vertex under density assumptions.

Findings

For large enough finite fields, many cycles of length 2k exist through each vertex.

The number of such cycles grows polynomially with the size of the field.

The result applies to subsets of the sphere in finite field vector spaces with positive density.

Abstract

Let $E$ be a proper symmetric subset of $S^{d-1}$, and $C_{\mathbb{F}_q^d}(E)$ be the Cayley graph with the vertex set $\mathbb{F}_q^d$, and two vertices $x$ and $y$ are connected by an edge if $x-y\in E$. Let $k\ge 2$ be a positive integer. We show that for any $\alpha\in (0, 1)$, there exists $q(\alpha, k)$ large enough such that if $E\subset S^{d-1}\subset \mathbb{F}_q^d$ with $|E|\ge \alpha q^{d-1}$ and $q\ge q(\alpha, k)$, then for each vertex $v$, there are at least $c(\alpha, k)q^{\frac{(2k-1)d-4k}{2}}$ cycles of length $2k$ with distinct vertices in $C_{\mathbb{F}_q^d}(E)$ containing $v$. This result is the inverse version of a recent result due to Iosevich, Jardine, and McDonald (2021).