Pinned distance sets, Wolff’s exponent in finite fields and improved sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that |∆(E)| & q whenever |E| & q, where E ⊂ Fq , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ∆(E) = {(x1 − y1) 2 + · · ·+ (xd − yd) 2 : x, y ∈ E}. The second listed author and Misha Rudnev ([4]) established the threshold d+1 2 , and in [3] the authors of this paper, Doowon Koh and Misha Rudnev proved that this exponent is sharp in even dimensions. In this paper we improve the threshold to d 2 2d−1 under the additional assumption that E has product structure. In particular, we obtain the exponent 4 3 , consistent with the corresponding exponent in Euclidean space obtained by Wolff ([9]).