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In recent years, sum-product estimates in Euclidean space and finite fieldshave been studied using a variety of combinatorial, number theoretic andanalytic methods. Erdos type problems involving the distribution of distances,areas and volumes have also received much attention. In this paper we prove arelatively straightforward function version of an incidence results for pointsand planes previously established in \cite{HI07} and \cite{HIKR07}. As aconsequence of our methods, we obtain sharp or near sharp results on thedistribution of volumes determined by subsets of vector spaces over finitefields and the associated arithmetic expressions.  In particular, our machinery enables us to prove that if $E \subset {\BbbF}_q^d$, $d \ge 4$, the $d$-dimensional vector space over a finite field ${\BbbF}_q$, of size much greater than $q^{\frac{d}{2}}$, and if $E$ is a productset, then the set of volumes of $d$-dimensional parallelepipeds determined by$E$ covers ${\Bbb F}_q$. This result is sharp as can be seen by taking $E$ toequal to $A \times A \times ... \times A$, where $A$ is a sub-field of ${\BbbF}_q$ of size $\sqrt{q}$. In three dimensions we establish the same result if$|E| \gtrsim q^{{15/8}}$. We prove in three dimensions that the set of volumescovers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^{{3/2}}$. Finallywe show that in three dimensions the set of volumes covers a positiveproportion of ${\Bbb F}_q$ if $|E| \ge Cq^2$, without any further assumptionson $E$, which is again sharp as taking $E$ to be a 2-plane through the originshows.

Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields