Research at the Interface of Harmonic Analysis and Arithmetic Combinatorics: Geometric Ramsey Theory and Higher Uniformity Norms

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This project deals with the occurrence of geometric patterns in sufficiently large but otherwise arbitrary sets, a field of mathematics most commonly referred to as (geometric) Ramsey theory. It is specifically focused on the study the question of determining whether or not such sets will be guaranteed to contain a translated and rotated copy of a given finite set, or of its sufficiently large dilates. This study has potential applications to analyzing and determining the true complexity of large data sets. Over the last twenty years, specifically after the groundbreaking work of Gowers on quantitative questions concerning the existence of arbitrarily long sequences of equally spaced numbers in any sufficiently large subsets of the integers, there has been remarkable progress in the study of general linear patterns via the development of so-called higher-order Fourier analysis. Perhaps the most notable achievement here is the celebrated result of Green and Tao on arbitrarily long sequences of equally spaced prime numbers. This project builds on these developments, and one of its major objectives is the development of analytic tools to understand the aforementioned occurrence of geometric and arithmetic structures in large sets. The problems under consideration arise in the context of both the integer lattice and classical Euclidean spaces. The principal investigator's approach blends the well-established delicate interplay between techniques from discrete harmonic analysis and number theory, with a new general approach based on the modern point of view of additive combinatorics.The existence of prescribed geometric structures in large subsets of the integer lattice and also in large measurable subsets of Euclidean spaces is currently not well understood. The project aims to address several such problems using the modern point of view of additive combinatorics. Specifically, the approach of showing that the count of isometric copies of a given finite configuration contained a given set is controlled by certain norms of its so-called balance function. These norms measure the uniformity or randomness of the set and, if it is sufficiently small with respect to the set's density, then the set will contain the expected number of isometric copies. The next step is to establish an inverse theorem showing that the largeness of the norm of the balance function implies that the set correlates or can be approximated by some structured object on which one can iterate this procedure. Recent results of the principal investigator with collaborators indicate that this should indeed the correct framework within which one should be approaching these problems and suggest that questions concerning geometric configurations of different levels of complexity can be tackled in a systematic way using appropriate higher (geometric) uniformity norms. The ultimate goal of this project is to characterizing those finite geometric configuration for which all its sufficiently large dilates can be realized in subsets of Euclidean space of positive upper density, strengthening existing connections between additive combinatorics and classical harmonic analysis, and establishing the analogous characterization in the discrete setting of the integer lattice.