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48
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
, 2010
"... The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a ..."
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The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a polynomial of degree at most d − 1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F> d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial φ is allowed to be of some larger degree C(d). The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture.
Ergodic seminorms for commuting transformations and applications. Available at http://fr.arxiv.org/abs/0811.3703
"... Abstract. Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. O ..."
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Abstract. Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide some tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they will be useful in the solution of other problems. 1.
Norm convergence of nilpotent ergodic averages
, 2012
"... Abstract We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L 2 norm. ..."
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Abstract We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L 2 norm.
Multiple recurrence for two commuting transformations. À paraître, Ergodic Theory Dynam. Systems
"... Abstract. This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of “magic systems ” established recentl ..."
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Abstract. This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of “magic systems ” established recently by B. Host for the proof. hal00441767, version 1 17 Dec 2009 1.
Deducing the density HalesJewett theorem from an infinitary removal lemma
 J. Theoret. Probab
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Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
 Pacific J. Math
"... Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, on ..."
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Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, one has lim infN→ ∞ 1N ∑N n=1 Re τ(aα n(a)... α(k−1)n(a))> 0; a subsequent result of Host and Kra shows that this limit exists. In particular, Re τ(aαn(a)... α(k−1)n(a))> 0 for all n in a set of positive density. From the von Neumann algebra perspective, it is thus natural to ask to what extent these results remain true when the abelian hypothesis is dropped. All three claims hold for k = 2, and we show in this paper that all three claims hold for all k when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for k = 3 when the von Neumann algebra is
ERGODIC AVERAGES OF COMMUTING TRANSFORMATIONS WITH DISTINCT DEGREE POLYNOMIAL ITERATES
, 2010
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