  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 $\Delta(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in
E\}$. In two dimensions we improve the known exponent to $\tfrac{4}{3}$,
consistent with the corresponding exponent in Euclidean space obtained by
Wolff.
  The pinned distance set $\Delta_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x\in
E\}$ for a pin $y\in E$ has been studied in the Euclidean setting. Peres and
Schlag showed that if the Hausdorff dimension of a set $E$ is greater than
$\tfrac{d+1}{2}$ then the Lebesgue measure of $\Delta_y(E)$ is positive for
almost every pin $y$. In this paper we obtain the analogous result in the
finite field setting. In addition, the same result is shown to be true for the
pinned dot product set $\Pi_y(E)=\{x\cdot y: x\in E\}$. Under the additional
assumption that the set $E$ has cartesian product structure we improve the
pinned threshold for both distances and dot products to $\frac{d^2}{2d-1}$.
  A generalization of the Falconer distance problem is determine the minimal
$\alpha>0$ such that $E$ contains a congruent copy of every $k$ dimensional
simplex whenever $|E| \gtrsim q^{\alpha}$. Here the authors improve on known
results (for $k>3$) using Fourier analytic methods, showing that $\alpha$ may
be taken to be $\frac{d+k}{2}$.
