Congruence classes of large configurations in vector spaces over finite fields

Abstract

Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such that any set $E\subset {\mathbb{F}}_q^d$ with $|E|\gtrsim q^s$ determines $\gtrsim q^{\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)}$ congruence classes of $(k+1)$-point configurations for $k\le d$. Because congruence classes can be identified with tuples of distances between distinct points when $k\le d$, and because there are $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ such pairs, this means any such $E$ determines a positive proportion of all congruence classes. In the $k>d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|\gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $k\le d$ case.