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

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).

16 pages. V2 with some small changes