Combinatorial Number Theory

University Of California-Riverside, Riverside CA

Investigators

Abstract

Freiman's Theorem says that given a set A, if the sum set (i.e. the set of sums of two elements in A) is not too big, then A can be included in the homomorphic image in the set of integers of a d-dimensional box with integer coordinates. We propose to sharpen our bound on the size of the box. We would also like to improve our results on some conjectures made by Erdos and Szemeredi in combinatorial number theory, which says that the sum set and product set cannot both be small. We would like to generalize some of the results to h-fold sum or product and sum-product problems along graphs. The importance of these and similar issues became more apparent in recent years because of their relation to issues in computer science and harmonic analysis. Important contributions along this line were made by Fefferman and Gowers. One of the features of several current developments in mathematics and applied mathematics is the emergence of issues with a combinatorial flavor, sometimes in seemingly unrelated questions. Some of these issues turned out to have already been studied in earlier days with different motivations. Well-known examples of this phenomenon is the revitalization of mathematical topics such as graph theory and computational algebra under impetus of computer science, in particular complexity theory and the theory of algorithms. My work in combinatorial number theory has this feature. But besides an interest from part of the computer science community, it does primarily belong to an active research direction in harmonic analysis and differential equations.