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 received great attention. They can often be interpreted in terms of Erdős type incidence problems involving the distribution of distances, dot products, areas, and so on, which have been studied quite extensively by way of combinatorial and Fourier analytic techniques. We use both kinds of techniques to obtain sharp or near-sharp results on the distribution of volumes   (as examples of dd-linear homogeneous forms) determined by sufficiently large subsets of vector spaces over finite fields and the associated arithmetic expressions. Arithmetic–combinatorial techniques turn out to be optimal for dimension d≥4d≥4 to this end, while for d=3d=3 they have failed to provide us with a result that follows from the analysis of exponential sums. To obtain the latter result we prove a relatively straightforward function version of an incidence results for points and planes previously established in [D. Hart, A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in: Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464 (2008); D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture, arXiv:math/0711.4427, preprint 2007].More specifically, we prove that if E=A×⋯×AE=A×⋯×A is a product set in Fqd, d≥4d≥4, the dd-dimensional vector space over a finite field FqFq, such that the size |E||E| of EE exceeds qd2 (i.e. the size of the generating set AA exceeds q) then the set of volumes of dd-dimensional parallelepipeds determined by EE covers FqFq. This result is sharp as can be seen by taking A=FpA=Fp, a prime sub-field of its quadratic extension FqFq, with q=p2q=p2. For in three dimensions, however, we are able to establish the same result only if |E|≳q158 (i.e., |A|≥Cq58, for some CC; in fact, the q158 bound can be justified for a slightly wider class of “Cartesian product-like” sets), and this uses Fourier methods. Yet we do prove a weaker near-optimal result in three dimensions: that the set of volumes generated by a product set E=A×A×AE=A×A×A covers a positive proportion of FqFq if |E|>q32 (so |A|>q). Besides, without any assumptions on the structure of EE, we show that in three dimensions the set of volumes covers a positive proportion of FqFq if |E|≥Cq2|E|≥Cq2, which is again sharp up to the constant CC, as taking EE to be a 2-plane through the origin shows.