From harmonic analysis to arithmetic combinatorics

Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably best-known result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the long-standing conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the Green-Tao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin. The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the Green-Tao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles. An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated. We will not try to give a comprehensive survey of harmonic analysis, combinatorics, or additive number theory. We will not even be able to do full justice to our specific areas of focus, instead referring the reader to the more complete expositions and surveys listed in Section 7. Our goal here is to emphasize the connections between these areas; we will thus concentrate on relatively few problems, chosen as much for their importance to their fields as for their links to each other. The article is written from the point of view of an analyst who, in the course of her work, was gradually introduced to the questions discussed here and found them fascinating. We hope that the reader will enjoy a taste of this experience.

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