A note on star-like configurations in finite settings

Abstract: Given $E \subset \mathbb{F}_q^d$, we show that certain configurations occur frequently when $E$ is of sufficiently large cardinality. Specifically, we show that we achieve the statistically number of $k$-stars $\displaystyle\left|\left{(x, x^1, \dots, x^k) \in E^{k+1} : | x - x^i | = t_i \right}\right|$ when is $|E| \gg_k q^{\frac{d+1}{2}}$. This result can be thought of as a natural generalization of the Erd\H os-Falconer distance problem. Our result improves on a pinned-version of our theorem which implied the above result, but only in the range $|E| \gg q^{\frac{d+k}{2}}$. As an immediate corollary, this demonstrates that when $|E| \gg c_k q^{\frac{d+1}{2}}$, then $E$ determines a positive proportion of all $k$-stars. Our results also extend to the setting of integers mod $q$.