We offer a proof of the following nonconventional ergodic theorem:The leading case of this result, with r = 1 and the standard sequence of averaging sets, was first proved by Tao in [16], following earlier analyses of various more special cases and related results by Conze and Lesigne [4,5,6], Furstenberg and Weiss [9], Zhang [18], Host and Kra [12,13], Frantzikinakis and Kra [7] and Ziegler [19]. While Tao's proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

Two versions of the Ajtai-Szemerédi Theorem are considered in the Cartesian square of a finite non-Abelian group G. In case G is sufficiently quasirandom, we obtain strong forms of both versions: if E ⊆ G × G is fairly dense, then E contains a large number of the desired patterns for most individual choices of 'common difference'. For one of the versions, we also show that this set of good common differences is syndetic.

Let pX, dq be a compact metric space and µ a Borel probability on X. For each N ě 1 let d N 8 be the ℓ8-product on X N of copies of d, and consider 1-Lipschitz functions X N ÝÑ R for d N 8 . If the support of µ is connected and locally connected, then all such functions are close in probability to juntas: that is, functions that depend on only a few coordinates of X N . This describes the failure of measure concentration for these product spaces, and can be seen as a Lipschitz-function counterpart of the celebrated result of Friedgut that Boolean functions with small influences are close to juntas.

Szemerédi's Theorem asserts that any positive-density subset of the integers must contain arbitrarily long arithmetic progressions. It is one of the central results of additive combinatorics. After Szemeredi's original combinatorial proof, Furstenberg noticed the equivalence of this result to a new phenomenon in ergodic theory that he called 'multiple recurrence'. Furstenberg then developed some quite general structural results about probabilitypreserving systems to prove the Multiple Recurrence Theorem directly. Furstenberg's ideas have since given rise to a large body of work around multiple recurrence and the associated 'non-conventional' ergodic averages, and to further connections with additive combinatorics. This course is an introduction to multiple recurrence and some of the ergodic theoretic structure that lies behind it. We begin by explaining the correspondence observed by Furstenberg, and then give an introduction to the necessary background from ergodic theory. We emphasize the formulation of multiple recurrence in terms of joinings of probability-preserving systems. The next step is a proof of Roth's Theorem (the first nontrivial case of Szemeredi's Theorem), which illustrates the general approach. We finish with a proof of a more recent convergence theorem for some non-conventional ergodic averages, showing some of the newer ideas in use in this area. The classic introduction to this area of combinatorics and ergodic theory is Furstenberg's book [Fur81], but the treatment below has a more modern point of view. 1.1 Szemerédi's Theorem and its relatives In 1927, van der Waerden gave a clever combinatorial proof of the following surprising fact: This easily implies Szemerédi's Theorem, because if k ≥ 1, E ⊆ Z has d(E) > 0, and we define Π : Z k−1 −→ Z : (a 1 , a 2 ,. .. , a k−1) → a 1 + 2a 2 + • • • + (k − 1)a k−1 , then the pre-image Π −1 (E) hasd(Π −1 (E)) > 0, and an upright isosceles simplex found in Π −1 (E) projects under Π to a k-term progression in E. Similarly, by projecting from higher-dimensions to lower, one can prove that Theorem 1.4 actually implies the following: Corollary 1.5 If F ⊂ Z d is finite and E ⊆ Z d hasd(E) > 0, then there are some a ∈ Z d and n ≥ 1 such that {a + nb : b ∈ F} ⊆ E. D R A F T 8 T. Austin rather than just homeomorphisms. However, general measurable spaces can exhibit certain pathologies which Borel σ-algebras of compact metrizable spaces cannot. The real assumption we need here is that our measurable spaces be 'standard Borel', but the assumption of a compact metric is a convenient way to guarantee this. Having explained this, beware that many authors restrict the convenient term 'G-space' to actions by homeomorphisms. 1.2.2 The phenomenon of multiple recurrence In order to introduce multiple recurrence, it is helpful to start with the probabilitypreserving version of Poincaré's classical Recurrence Theorem. Theorem 1.6 (Poincaré Recurrence) If (X, µ, T) is a system and A ∈ B X has µ(A) > 0, then there is some n = 0 such that µ(A ∩ T −n A) > 0. Proof The pre-imag...