Abstract. There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms U d (G), d = 1, 2, 3,... on a finite additive group G; in particular, to detect arithmetic progressions of length k in G it is important to know under what circumstances the U k−1 (G) norm can be large. The U 1 (G) norm is trivial, and the U 2 (G) norm can be easily described in terms of the Fourier transform. In this paper we systematically study the U 3 (G) norm, defined for any function f: G → C on a finite additive group G by the formula

Abstract. A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory. 1. Background For hundreds of years, mathematicians have made conjectures about patterns in the primes: one of the simplest to state is that the primes contain arbitrarily long arithmetic progressions. It is not clear exactly when this conjecture was first formalized, but as early as 1770 Lagrange and Waring studied the problem of how large the common difference of an arithmetic progression of k primes must be. A

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Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {p, 2p,..., kp}. We then derive several combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all ε> 0 and every subset of the integers Λ the set n ∈ N: d ∗ ( Λ ∩ (Λ + p1(n)) ∩ (Λ + p2(n)) ∩ (Λ + p3(n)) )> (d ∗ (Λ)) 4 − ε} has bounded gaps for “most ” choices of integer polynomials p1, p2, p3. Contents

Abstract. For any measure preserving system (X, X, µ, T) and A ∈ X with µ(A)> 0, we show that there exist infinitely many primes p such that µ ` A ∩ T −(p−1) A ∩ T −2(p−1) A ´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p. 1.

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3 Abstract 3 Chapter 1. Introduction 7 1. Correlation Sequences 7 2. The Kronecker Factor 9 3. Generalized Correlation Sequences 12 Chapter 2. Useful Facts 15 1. Conditional Expectation and Disintegration of Measures 15 2. Group Extensions 18 3. Product Measures and Conditional Product Measures 20 4. Nilpotent Systems 23 Chapter 3. GCS and Nilpotent systems 25 1. Preliminaries 25 2. The Ergodic Decomposition of T \Theta T \Theta T 2 \Theta T 2 29 3. From Lesigne Equation to Nil-Systems 40 Chapter 4. The Weyl Algebra Does Not Include All GCS 47 1. The Heisenberg Manifold 47 2. Trigonometric Polynomials in Ergodic Systems 51 3. Constructing a GCS Orthogonal to the Weyl Algebra 52 References 63 2 CONTENTS Abstract Let (X; B; ¯; T ) be a measure preserving system, i.e. (X; B;¯) is a probability space and T : X \Gamma! X a measurable and measure preserving map. We assume that the system is ergodic: T \Gamma1 A = A ) ¯(A) = 0 or ¯(A) = 1. In order to investigate the way T "mixes"...

Abstract. Let X = (X 0, B, µ, T) be an ergodic probability measure preserving system. For a natural number k we consider the averages N ∑ k ∏ 1 fj(T

Abstract. Let X = (X 0, B, µ, T) be an ergodic probability measure preserving system. For a natural number k we consider the averages N ∑ k ∏ 1 (1) fj(T

Abstract. We study equidistribution properties of nil-orbits (b n x)n∈N when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if X = G/Γ is a nilmanifold, b ∈ G is an ergodic nilrotation, and c ∈ R \ Z is positive, then the sequence (b [nc] x)n∈N is equidistributed in X for every x ∈ X. This is also the case when n c is replaced with a(n), where a(t) is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when X is the circle. Contents