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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 famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.

Abstract. We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1.

The problem of constructing dense subsets S of {1, 2,..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with |S | = Ω(nlog3 2) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is

Let r(k, N) be the maximal cardinality of a subset A of {1, 2,..., N} which does not contain an arithmetic progression of length k. That is, A does not contain a subset of the form {x + jy: 0 ≤ j < k}, where x, y are integers with y ̸ = 0. Erdös and Turan [3] initiated the study of these quantities in 1936. In particular they conjectured that r(k, N) = o(N) for all k, that is every set of integers of positive asymptotic density contains arbitrarily long arithmetic progressions. In 1953, Roth [8] showed that r(3, N) = o(N). The Erdös–Turan conjecture was verified by Szemerédi [11, 12], a result with a very broad influence. Subsequently, rather different proofs of Szemerédi’s theorem were given by Furstenberg [4] and Gowers [5, 6]. Gowers’s proof provides, for the first time, upper bounds on r(k, N) given by a bounded tower of exponentials. An intriguing question of Erdös asks if r(3, N) ≤ CN/(log N) 1+δ for some positive δ. Bourgain’s article [2] contains the best current upper bound of CN √ log log N on r(3, N). log N In this article we are interested in the converse question of finding large subsets of {1,..., N} which do not contain arithmetic progressions. Behrend, in 1946, [1] (building on earlier work of Salem and Spencer [10]) considered three term arithmetic progressions, and showed that r(3, N) ≥ N exp(−C √ log N). The purpose of this paper is to show that if one considers longer arithmetic progressions then Behrend’s estimate can be further improved as follows.

For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H) if it contains no copy (resp. induced copy) of D. Our goal in satisfies property PD (resp. P ∗ D this paper is to classify the k-graphs D for which there are property-testers for testing PD and P ∗ D whose query complexity is polynomial in 1/ɛ. For such k-graphs we say that PD (resp. P ∗ D) is easily testable. For P ∗ D, we prove that aside from a single 3-graph, P ∗ D is easily testable if and only if D is a single k-edge. We further show that for large k, one can use more sophisticated techniques in order to obtain better lower bounds for any large enough k-graph. These results extend and improve previous results about graphs [5] and k-graphs [18]. For PD, we show that for any k-partite k-graph D, PD is easily testable, by giving an efficient one-sided error-property tester, which improves the one obtained by [18]. We further prove a nearly matching lower bound on the query complexity of such a property-tester. Finally, we give a sufficient condition for inferring that PD is not easily testable. Though our results do not supply a complete characterization of the k-graphs for which PD is easily testable, they are a natural

Abstract. Let k � 3 be an integer, and let G be a finite abelian group with |G | = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality |A | of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(Z/NZ) = ok(N). It is known, in fact, that the estimate rk(G) = ok(N) holds for all G. There have been many papers concerning the issue of finding quantitative bounds for rk(G). A result of Bourgain states that r3(G) ≪ N(log log N / logN) 1/2 for all G. In this paper we obtain a similar bound for r4(G) in the particular case G = F n, where F is a fixed finite field with char(F) ̸ = 2, 3 (for example, F = F5). We prove that r4(G) ≪F N(log N) −c for some absolute constant c> 0. In future papers we will treat general abelian groups G, eventually obtaining a comparable result for arbitrary G. 1.

For any coloring of the N × N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d).