Group actions and geometric combinatorics in F^d/_q

In this paper we apply a group action approach to the study of Erdős–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 s0(d)<d such that if E⊂Fdq, d≥2, with |E|≥Cqs0, then |Tdd(E)|≥C′q(d+12), where Tdk(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 [4] for Tdk(E) with 2≤k≤d−1. A non-trivial result for T22(E) in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 43, previously established in [4] for the q≡3mod4 case to the case q≡1mod4, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in Fdq, 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.