Arithmetic Combinatorics and Applications to Number Theory

University Of California-Riverside, Riverside CA

Investigators

Abstract

This research project concerns arithmetic combinatorics, an interdisciplinary field of research with many emerging applications. Progress on questions in number theory and theoretical computer science requires new insights and methods; part of the success in this direction over recent years relies on novel techniques at the interface of algebra and combinatorics. Central to these developments is the so-called arithmetic combinatorics of finite fields, which has undergone significant recent advances and opened new challenges. This research project focuses on several questions motivated by current research in the area, including study of the orders of points on varieties and counting solutions to algebraic equations with constraint variables. Combinatorial problems in finite fields continue to offer many challenges. Of particular interest in this research project are questions involving orders of points on varieties over finite fields (e.g. recent developments related to the Markoff surface). Part of the motivation for the work is to investigate the problem of strong approximation for Markoff triples and to give estimates on the number of solutions of equations when the variables are restricted one way or another, for instance to multiplicative groups. When classical techniques do not apply, general sum product theory in finite fields and residue rings may be useful. Sum-product results in various settings are of interest in their own right as they lead to new results in analytic number theory, in particular, estimates on Gauss sums and short character sums.