In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in [10] and [12]. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions.In particular, our machinery enables us to prove that if E ⊂ F d q , d ≥ 4, the ddimensional vector space over a finite field F q , of size much greater than q d 2 , and if E is a product set, then the set of volumes of d-dimensional parallelepipeds determined by E covers F q . This result is sharp as can be seen by taking E to equal to A × A × · · · × A, where A is a sub-field of F q of size √ q. In three dimensions we establish the same result if |E| q 15 8 . We prove in three dimensions that the set of volumes covers a positive proportion of F q if |E| ≥ Cq 3 2 . Finally we show that in three dimensions the set of volumes covers a positive proportion of F q if |E| ≥ Cq 2 , without any further assumptions on E, which is again sharp as taking E to be a 2-plane through the origin shows.

We study variants of the Erdős distance problem and the dot products problem in the setting of the integers modulo q, where q = p ℓ is a power of an odd prime.

Abstract. Let E ⊂ F d q , the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y ∈ E by an edge if ||x − y|| := (We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, 1 · |E| k+1 q −k plus a much smaller remainder.

Abstract. For a set E ⊂ F d q , we define the k-resultant magnitude set asIn this paper we find a connection between a lower bound of the cardinality of the k-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if +ε for ε > 0, then |∆ 3 (E)| ≥ cq.

Abstract. We study the k-resultant modulus set problem in the ddimensional vector space F d q over the finite field Fq with q elements.