Group actions and geometric combinatorics in IFqd

In this paper we apply a group action approach to the study of Erdos-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 of F-q(d), d >= 2, with vertical bar E vertical bar >= Cq(s0), then vertical bar T-d(d) (E)vertical bar >= C'q((d+1)(2)), where T-k(d)(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 T-k(d)(E) with 2 <= k <= d - 1. A non-trivial result for T-2(2)(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 4/3, previously established in [4] for the q equivalent to 3 mod 4 case to the case q equivalent to 1 mod 4, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in 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.