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34
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
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Cited by 81 (4 self)
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Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
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A new proof of the density HalesJewett theorem
, 2009
"... Abstract The HalesJewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1, . . . , k} n contains a monochromatic combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundament ..."
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Cited by 39 (2 self)
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Abstract The HalesJewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1, . . . , k} n contains a monochromatic combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdős and Turán in 1936, proved by Szemerédi in 1975, and given a different proof by Furstenberg in 1977. The HalesJewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemerédi's theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of {1, 2, 3} n of density δ contains a combinatorial line if n is at least as big as a tower of 2s of height O(1/δ 2 ). Our proof is surprisingly simple: indeed, it gives arguably the simplest known proof of Szemerédi's theorem.
The ergodic and combinatorial approaches to Szemerédi’s theorem
, 2006
"... A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approach of Szem ..."
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Cited by 17 (2 self)
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A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourieranalytic approach of Gowers, and the hypergraph approach of NagleRödlSchachtSkokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.
Central sets and a noncommutative Roth theorem
 American J. Math
"... Abstract. In 1952 K. Roth showed that any subset of N having positive upper density contains arithmetic progressions of length 3, a result E. Szemerédi extended to progressions of arbitrary length in 1975. An approach involving multiple recurrence for measure preserving systems developed by H. Fur ..."
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Cited by 15 (6 self)
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Abstract. In 1952 K. Roth showed that any subset of N having positive upper density contains arithmetic progressions of length 3, a result E. Szemerédi extended to progressions of arbitrary length in 1975. An approach involving multiple recurrence for measure preserving systems developed by H. Furstenberg led to extensions to linear configurations in more general groups while demonstrating families of such configurations to be large in various senses. Using ultrafilter techniques for doing ergodic theory without averaging, we prove a version of Roth’s theorem, in its ergodictheoretic formulation, for general countable groups G. A combinatorial consequence, valid in amenable groups, is the presence of many configurations of the form {(a, b), (ag, b), (ag, bg)} in positive density subsets of G × G. Here “many ” implies in particular “central * in the variable g”. 1. Introduction. K. Roth
Deducing the density HalesJewett theorem from an infinitary removal lemma
 J. Theoret. Probab
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A simple regularization of hypergraphs
, 2009
"... We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its mult ..."
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Cited by 6 (3 self)
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We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by FurstenbergKatznelson (1978).