This is a proposal in arithmetic combinatorics, which has become an interdisciplinary field of research with many applications. While certain themes in additive combinatorics are classical in number theory, there is also focus on new structural questions that turned out to be important. For instance, the work of Gowers and, later, Green and Tao on arithmetic progressions have put considerable impetus on Freiman's theorem and it's quantitative versions. Parallel to these results, a general `sum-product' theory in finite fields and residue rings was developed. The roots of this research go back for instance to early work of Erdos-Szemeredi and the finite field version of the Kakeya problem (solved by Z. Dvir). It turned out that results from sum-product and product theory in various settings are of interest in their own right as they lead to new results in analytic number theory (such as estimates of short character sums), in pseudo-randomness and in group theory (growth, expansion and spectral gaps). The PI intends to explore finite field analogues of the Szemeredi Trotter theorem on incidences for algebraic curves (which are a special case of so-called pseudo-line systems). In the finite field setting, such results are presently only available for straight lines. More general results for pseudo-line systems have been obtained over the reals but none of the known approaches seem adaptable to the finite fields situation, so that new ideas are clearly needed here. Results of this type would have major implications in the areas mentioned above because they allow to obtain nontrivial statements on solutions of systems of equations in situations where classical number theory is not applicable. As a particular case, the PI would like to obtain analogues of the Bombieri-Pila results on lattice points on curves restricted to boxes in the prime field setting. Related to 'growth' phenomena in groups, the work of Breuillard, Green and Tao provides a complete description of 'approximate groups' that in some sense generalize Freiman's theorem and also provide a finitary version of Gromov's theorem. At this stage, the results are only qualitative and obtaining quantitative versions would be most interesting, in particular in view of the consequences to group expansion. The PI will continue to work with her collaborators on Poonen's conjecture on the multiplicative order of F-points on curves. Again this is a problem at the interface of combinatorics, algebra and number theory where progress can be expected.
Arithmetic combinatorics and `sum-product theory' in various settings have become increasingly significant to various other fields, such as pseudo-randomness in computer science, classical analytic number theory and the theory of expansion in linear groups. The purpose of this proposal is to continue research on the related issues in combinatorial number theory and their applications in particular, to problems of estimating the number of solutions of algebraic equations when the variables are restricted. This research involves different groups of people and the interaction of various branches of mathematics, occasionally leading to progress on old problems.
