Some problems at the interface of harmonic analysis, number theory, and combinatorics

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

So-called Ramsey theory deals with the problem of finding structures in large but otherwise disorganized sets. In the geometric setting it is to show that such sets contain a translated and rotated copy of a given finite set, or of its sufficiently large dilates. In other words it is to study the occurrence of geometric patterns. Over the past fifteen years there has been a remarkable progress of the study of linear patterns, developing and introducing tools from mathematical analysis, often referred to as higher-order Fourier analysis. Among the major achievements is the celebrated result of Green and Tao, which states that there are arbitrary long sequences of equally spaced prime numbers. This project builds on this development, and one of its major objectives is to develop analytic tools to understand the occurrence of geometric and arithmetic (i.e., defined by equations) structures in large but otherwise arbitrary sets. The problems arise in the context of the prime and integer lattice and also in classical Euclidean spaces. The principal investigator's approaches involve the interplay of techniques from discrete harmonic analysis and number theory, in addition to a new ingredient, ideas from additive combinatorics. The first motivational context for the project is that of prime numbers: to study nonlinear relations among the primes and to investigate the related problem of finding geometric constellations among points with prime coordinates. The underlying philosophy of considering the primes as a random subset of the integers leads naturally to the study of analogous questions in large sets of integer points and also in large measurable subsets of Euclidean spaces. Geometric structures in such sets are not well understood. The project aims to develop a general approach based on the modern point of view of additive combinatorics; namely, to establish appropriate notions of randomness that control the frequency at which a certain pattern occurs and to prove structure theorems for sets that are not suitably random. The underlying constructs are analytic and are related to objects studied in discrete harmonic analysis such as maximal operators and Radon transforms acting on functions defined on the integer lattice. Finally, the project aims to study geometric patterns in large measurable subsets of Euclidean spaces from this novel point of view, strengthening the connections between additive combinatorics and classical harmonic analysis.