Results 1  10
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25
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 75 (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 15 (5 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. 0.1. The celebrated van der Waerden Theorem ([W]) states that if the
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 12 (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.
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
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 5 (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.”
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.
Minimal Invariant Spaces in Formal Topology
 The Journal of Symbolic Logic
, 1996
"... this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a ..."
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Cited by 4 (1 self)
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this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace
Extraction of bounds: Interpreting some tricks of the trade. In P. Suppes (Ed.), Universitylevel computerassisted instruction at
 Stanford University, Institute for
, 1978
"... THIS ARTICLE is an addendum to my other piece in this volume (Kreisel, 1981), NP for short. Section I below is principally a summary of recent mathematical progress on two topics raised in NP, but formal details will be published separately. Section 2 goes into the need for new questions to cope wit ..."
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Cited by 4 (0 self)
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THIS ARTICLE is an addendum to my other piece in this volume (Kreisel, 1981), NP for short. Section I below is principally a summary of recent mathematical progress on two topics raised in NP, but formal details will be published separately. Section 2 goes into the need for new questions to cope with the law of diminishing returns. Summary. Section 1 elaborates the following point which appeared in NP only in side remarks, especially in the footnotes to sections 3.3 and 4.5. Mathematically trivial changes in formalizations of a given (nonconstructive) proof can affect significantly the prooftheoretic transformations applied to the formalizations, and, in the case of II~theorems, even the bounds extracted by these transformations. This situation is in sharp contrast to the results reported in section 3.1 of NP (stage 2), on the stability of bounds obtained from a given (intuitionistic) formalization by means of different transformations, but in accord with familiar experience sin<;:e, for our ordinary mathematicaljudgment, a proof
A constructive topological proof of van der Waerden's theorem
 Journal of Pure and Applied Algebra
, 1993
"... this paper was written, we became aware of the work [2, 15], which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic nonconstructive results, which become then constructively valid. Our work su ..."
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Cited by 3 (0 self)
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this paper was written, we became aware of the work [2, 15], which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic nonconstructive results, which become then constructively valid. Our work suggests that these methods, beside the purely mathematical advantage of solving problems concerned with equivariance or continuity in parameters [15], may be interesting also prooftheoretically in providing an elegant framework for extracting computational informations from given mathematical arguments. Our treatment of the topological proof of van der Waerden's theorem is apparently different from the one presented in Girard's book on proof theory [10]. We have not tried though to compare in detail the two arguments, because the main emphasis is somewhat different. The main points of our paper are the formulation of a pointfree version of a minimal property, whose ordinary version is proved via Zorn's lemma, and the observation that this pointfree version has a direct inductive proof