A good universal weight for nonconventional ergodic averages in norm

Assani, Idris; Moore, Ryo

doi:10.1017/etds.2015.76

Cited by 4 publications

(3 citation statements)

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“…where ||| · ||| µ,k is the seminorm on L ∞ (µ) defined in [46] in the ergodic case and in [20] in the general case. 3 We recall the definition and some properties of these seminorms in Appendix A. Note also that if ψ : N d → C is a nilsequence of the form (Φ(T n x)) n∈N d , then ψ admits correlations along every Følner sequence I and…”

Section: Uniformity Seminorms and Decomposition Resultsmentioning

confidence: 99%

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Weighted multiple ergodic averages and correlation sequences

Frantzikinakis¹

2016

Ergod. Th. Dynam. Sys.

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We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the averages of this sequence times any nilsequence converge. Key role in the proof play two decomposition results of independent interest. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and B. Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use the previous results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.Comment: 53 pages, small changes made in light of comments from the referee, to appear in Ergodic Theory and Dynamical System

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Section: Uniformity Seminorms and Decomposition Resultsmentioning

confidence: 99%

“…When d = 1, examples of good universal weights for some multiple ergodic averages can be found in [1,2,3,4,19,26,30,32,48,70]. Most of these results deal with the case where ℓ = 1 and are based on the theory of characteristic factors that was pioneered by H. Furstenberg.…”

Section: Introductionmentioning

confidence: 99%

Weighted multiple ergodic averages and correlation sequences

Frantzikinakis¹

2016

Ergod. Th. Dynam. Sys.

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“…For instance, the authors showed that the sequence a n = f 1 (T an x) f 2 (T bn x) is a good universal weight for the Furstenberg averages in the L 2 -norm for µ-a.e. x ∈ X [7], which was extended further to the case where we have commuting transformations [8]. The first author extended the double recurrence Wiener-Wintner result to nilsequences [3], and a result similar to this was announced by P. Zorin-Kranich independently [22].…”

Section: Introductionmentioning

confidence: 99%