# Congruence classes of large configurations in vector spaces over finite fields

## TL;DR

This paper extends previous results on the number of congruence classes determined by large sets in vector spaces over finite fields, establishing the correct count for configurations when the number of points exceeds the dimension.

## Contribution

It determines the exact number of congruence classes for large point sets in finite field vector spaces, especially when the number of points exceeds the space dimension.

## Key findings

- For $k
eq d$, the number of congruence classes is correctly counted as $q^{inom{k+1}{2}}$.
- Large sets with size $|E| extgreater q^s$ determine a positive proportion of all congruence classes.
- The results unify the understanding of configuration counts for all $k$, including overdetermined cases.

## 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 $\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations for $k\leq d$. Because congruence classes can be identified with tuples of distances between distinct points when $k\leq d$, and because there are $\binom{k+1}{2}$ 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 $q^{\binom{k+1}{2}}$ 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 $k\leq d$ case.