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 $\alpha\in (0, 1)$, 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).