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

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 $\alpha\in (0, 1)$,thereexists$q(\alpha, k)$largeenoughsuchthatif$E\subset S^{d-1}\subset $\mathbb{F}$q^d$with$|E|\ge \alpha q^{d-1}$and$q\ge q(\alpha, k)$,thenforeachvertex$v$,thereareatleast$c(\alpha, k)q^{\frac{(2k-1)d-4k}{2}}$cyclesoflength$2k$withdistinctverticesin$C{$\mathbb{F}$_q^d}(E)$containing$v$. This result is the inverse version of a recent result due to Iosevich, Jardine, and McDonald (2021).