  In this paper, we study the distribution of cycles in local Cayley distance
graphs. More precisely, 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 $v$. The novelty in our approach is a
connection to a recent result on $2k$-spherical configurations spanning $2k-2$
dimensions due to Lyall, Magyar, and Parshall (2020).
