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

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 cite{HI07} and cite{HIKR07}. 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 $E subset {Bbb F}_q^d$, $d ge 4$, the $d$-dimensional vector space over a finite field ${Bbb F}_q$, of size much greater than $q^{frac{d}{2}}$, and if $E$ is a product set, 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 $E$ to equal to $A times A times ... times A$, where $A$ is a sub-field of ${Bbb F}_q$ 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 volumes covers a positive proportion of ${Bbb F}_q$ if $|E| ge Cq^{{3/2}}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${Bbb F}_q$ if $|E| ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${mathbb F}_q$, the finite field with q elements, by $A cdot A+... +A cdot A$, where A is a subset ${mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.