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41
An ergodic Szemer'edi theorem for commuting transformations
 J. Analyse Math
, 1979
"... The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n> 0,/z (T'A n A)> 0. In [1] this was extended to multiple recurrence: ..."
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Cited by 114 (2 self)
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The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n> 0,/z (T'A n A)> 0. In [1] this was extended to multiple recurrence: the transformations T, T2,..., T k have a common power satisfying /x (A n ThA n... n Tk"A)> 0 for a set A of positive measure. We also showed that this result implies Szemer6di's theorem stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In [2] a topological analogue of this is proved: if T is a homeomorphism of a compact metric space X, for any e>0 and k = 1,2,3,.., there is a point x E X and a common power of T, T 2, 9 9 9 T k such that d(x, Tnx) < e, d(x, T2"x) < e,. 9 d(x, Tk~x) < e. This (weaker) result, in turn, implies van der Waerden's theorem on arithmetic progressions for partitions of the integers. Now in this case a virtually identical argument shows that the topological result is true for any k commuting transformations. This would lead one to expect that the measure theoretic result is also true for arbitrary commuting transformations. (It is easy to give a counterexample with noncommuting transformations.) We prove this in what follows. Theorem A. Let (X,~,/z) be a measure space with /z(X)<oo, let T~, T2, 9 9 9 Tk be commuting measure preserving transformations of X and let A E B with tz (A)> O. Then lim inf 1 N ~N 1 Iz ( T~'A A Tj~A N...A T~A)>O. A corollary is the multidimensional extension of Szemer6di's theorem: Theorem B. Let S C Z " be a subset with positive upper density and let F C Z " be any finite configuration. Then there exists an integer d and a vector n E Z " such that n+dFCS.
Setpolynomials and polynomial extension of the HalesJewett theorem
 ANN. OF MATH
, 1999
"... An abstract, HalesJewett type extension of the polynomial van der Waerden Theorem [BL] is established: Theorem. Let r,d,q ∈ N. There exists N ∈ N such that for any rcoloring of the set of subsets of V = {1,...,N} d × {1,...,q} there exist a set a ⊂ V and a nonempty set γ ⊆ {1,...,N} such that a ∩ ..."
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Cited by 20 (6 self)
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An abstract, HalesJewett type extension of the polynomial van der Waerden Theorem [BL] is established: Theorem. Let r,d,q ∈ N. There exists N ∈ N such that for any rcoloring of the set of subsets of V = {1,...,N} d × {1,...,q} there exist a set a ⊂ V and a nonempty set γ ⊆ {1,...,N} such that a ∩ (γ d × {1,...,q}) = ∅, and the subsets a, a ∪ (γ d × {1}), a ∪ (γ d × {2}),..., a ∪ (γ d × {q}) are all of the same color. This “polynomial ” HalesJewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of setpolynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.
The ergodic and combinatorial approaches to Szemerédi’s theorem
, 2006
"... Abstract. 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) approac ..."
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Cited by 17 (2 self)
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Abstract. 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. 1.
The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
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Cited by 13 (1 self)
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The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
Classifying dynamical systems by their recurrence properties
 Topol. Methods Nonlinear Anal
"... Abstract. In his seminal paper of 1967 on disjointness in topological dynamics and ergodic theory H. Furstenberg started a systematic study of transitive dynamical systems. In recent years this work served as a basis for a broad classification of dynamical systems by their recurrence properties. In ..."
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Cited by 13 (1 self)
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Abstract. In his seminal paper of 1967 on disjointness in topological dynamics and ergodic theory H. Furstenberg started a systematic study of transitive dynamical systems. In recent years this work served as a basis for a broad classification of dynamical systems by their recurrence properties. In this paper I describe some aspects of this new theory and its connections with combinatorics, harmonic analysis and the theory of topological groups. Contents
Kreisel's `Unwinding Program
 In Odifreddi [53
, 1996
"... Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give ..."
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Cited by 11 (0 self)
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Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give
ULTRAFILTERS: WHERE TOPOLOGICAL DYNAMICS = ALGEBRA = COMBINATORICS
, 1993
"... We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the HalesJewett partition theorem. ..."
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Cited by 7 (0 self)
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We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the HalesJewett partition theorem.
Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
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Cited by 5 (1 self)
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Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.