Group actions and geometric combinatorics in ${\mathbb F}_q^d$

In this paper we apply a group action approach to the study of Erd\H os-Falconer type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists $s_0(d)<d$ such that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, with $|E| \ge Cq^{s_0}$, then $|T^d_d(E)| \ge C'q^{d+1 \choose 2}$, where $T^d_k(E)$ denotes the set of congruence classes of $k$-dimensional simplices determined by $k+1$-tuples of points from $E$. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh (\cite{CEHIK12}) for $T^d_k(E)$ with $2 \leq k \leq d-1$. A non-trivial result for $T^2_2(E)$ in the plane was obtained by Bennett, Iosevich and Pakianathan (\cite{BIP12}). These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent $\frac{4}{3}$, previously established in \cite{CEHIK12} for the $q\equiv3\mbox{ mod }4$ case to the case $q\equiv1\mbox{ mod }4$, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in ${\mathbb F}_q^d$, where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.