Congruence classes of large configurations in vector spaces over finite fields
Alex McDonald · Functiones et Approximatio Commentarii Mathematici · 2019
In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s d$ 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.