  An analog of the Falconer distance problem in vector spaces over finite
fields asks for the threshold $\alpha>0$ such that $|\Delta(E)| \gtrsim q$
whenever $|E| \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the
$d$-dimensional vector space over a finite field with $q$ elements (not
necessarily prime). Here $\Delta(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in
E\}$. The second listed author and Misha Rudnev established the threshold
$\frac{d+1}{2}$, and 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 $\frac{d^2}{2d-1}$ 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.
