Arithmetic Combinatorics and Applications

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The research area of the project is arithmetic combinatorics. This has become an interdisciplinary field of research with many new emerging applications to analytic number theory, Fourier analysis, probability, computer science (e.g. elliptic curve cryptography, and pseudorandom number generators), and even theoretical physics (e.g. solid state physics). While certain themes in arithmetic combinatorics are classical in number theory, there is also focus on new structural questions that turned out to be important. Some of the useful tools (e.g. exponential sum and character sum estimates) are themselves attractive to researchers. Combinatorial problems in finite fields continue to offer many challenges, in particular questions involving orders of points on varieties over finite fields (e.g. recent developments related to the Markoff surface). This research involves different groups of people and the interaction of various branches of mathematics, with the potential of making progress on some well-known unsolved problems. The principal investigator will run a weekly seminar at graduate level on basic methods in analysis and combinatorics through explanation of research in combinatorial number theory. Special efforts would be made to attract first-generation college students.The goal of this project is to continue research on universality for nodal intersections, arithmetic progressions in multiplicative groups of finite fields, non-linear Roth type theorem, orders and density of points, on varieties over finite fields, and the theory of incomplete and short character sums. Besides the usual tools in (discrete) Fourier analysis and probability, techniques from arithmetic combinatorics were used, particularly, various versions of factorization in generalized arithmetic progressions and quantitative Nullstellensatz, sum-product theory for the exponential sum and character sum estimates, (including mixed character sums, character sums over very short intervals, and these sums with various moduli or arguments). It turns out that results from sum-product in various settings are of interest in their own right as well as they lead to new results in analytic number theory and some of the problems proposed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.