Abstract. This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next two lectures we prove two of the key background results, the

Abstract. The main goal of this survey is the description of the fruitful interaction between Ergodic Theory and Number Theory via the study of two beautiful results: the first one by Ben Green and Terence Tao (about long arithmetic progressions of primes) and the second one by Noam Elkies and Curtis McMullen (about the distribution of the sequence { √ n} mod 1). More precisely, during the first part, we’ll see how the ergodic-theoretical ideas of Furstenberg about the famous Szemerédi theorem were greatly generalized by Green and Tao in order to solve the classical problem of finding arbitrarily long arithmetical progression of prime numbers, while the second part will focus on how Elkies and McMullen used the ideas of Ratner’s theory (about the classification of ergodic measures related to unipotent dynamics) to compute explicitly the distribution of the sequence { √ n} on the unit circle.

The term additive combinatorics was coined a few years ago by Terry Tao to describe a rapidly developing and rather exciting area of mathematics. My personal experience is that rather few people have heard the term, though they are often familiar with some of the landmark results. When asked to define the area, I often experience a little difficulty, and in this respect I perhaps have a little in common with Dr. M. Kirschner, head of the Harvard School of Systems Biology, who said, 1 “Systems biology is like the old definition of pornography: I don’t know what it is, but I know it when I see it. ” He went on to say that “it’s a marriage of the natural science [sic] and computer science with biology, to try and understand complex systems. ” Well one might say that additive combinatorics is a marriage of number theory, harmonic analysis, combinatorics, and ideas from ergodic theory, which aims to understand very simple systems: the operations of addition and multiplication and how they interact. Even that definition is something of an oversimplification, as a glance at the choice of topics in the book under review shows. Let us begin by mentioning a

The term additive combinatorics was coined a few years ago by Terry Tao to describe a rapidly developing and rather exciting area of mathematics. My personal experience is that rather few people have heard the term, though they are often familiar with some of the landmark results. When asked to define the area, I often experience a little difficulty, and in this respect I perhaps have a little in common with Dr. M. Kirschner, head of the Harvard School of Systems Biology, who said, 1 “Systems biology is like the old definition of pornography: I don’t know what it is, but I know it when I see it. ” He went on to say that “it’s a marriage of the natural science [sic] and computer science with biology, to try and understand complex systems. ” Well one might say that additive combinatorics is a marriage of number theory, harmonic analysis, combinatorics, and ideas from ergodic theory, which aims to understand very simple systems: the operations of addition and multiplication and how they interact. Even that definition is something of an oversimplification, as a glance at the choice of topics in the book under review shows. Let us begin by mentioning a

neil lyall

A set A ⊆ N is square-difference free (henceforth SDF) if there do not exist x, y ∈ A, x ̸ = y, such that |x − y | is a square. Let sdf(n) be the size of the largest SDF subset of {1,..., n}. Ruzsa [10] has shown that proved sdf(n) ≥ Ω(n log 65 7) ≥ Ω(n 0.733077·· ·). sdf(n) = Ω(n 0.5(1+log 65 7) ) = Ω(n 0.733077·· ·) We improve on the lower bound by showing sdf(n) = Ω(n 0.5(1+log 205 12) ) = Ω(n 0.7334·· ·) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers. We also give the context and history of results of this type. 1

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