TY - JOUR

T1 - Multiple recurrence and nilsequences

AU - Bergelson, Vitaly

AU - Host, Bernard

AU - Kra, Bryna

AU - Ruzsa, Imre

N1 - Copyright:
Copyright 2005 Elsevier B.V., All rights reserved.

PY - 2005/5

Y1 - 2005/5

N2 - Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving system (X,χμT) and a set A ∈ χ of positive measure, the set of integers n such that μ (A ∩ Tn A ∩ T2n ... ∪ T kn A) > μ (A)k+1 - ∈ is syndetic. The size of this set, surprisingly enough, depends on the length (k + 1) of the arithmetic progression under consideration. In an ergodic system, for k = 2 and k = 3, this set is syndetic, while for k ≥ 4 it is not. The main tool is a decomposition result for the multicorrelation sequence ∫ f(x)f(Tnx) f(T 2nx) ... f(Tknx) dμ (x), where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d * (E) > 0 and for all ∈ ;> 0, the set {n ∈ ℤ: d* (E ∪ ( E + n) ∪ (E + 2n) ∪ (E + 3n)) > d* (E)4 - ∈} is syndetic.

AB - Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving system (X,χμT) and a set A ∈ χ of positive measure, the set of integers n such that μ (A ∩ Tn A ∩ T2n ... ∪ T kn A) > μ (A)k+1 - ∈ is syndetic. The size of this set, surprisingly enough, depends on the length (k + 1) of the arithmetic progression under consideration. In an ergodic system, for k = 2 and k = 3, this set is syndetic, while for k ≥ 4 it is not. The main tool is a decomposition result for the multicorrelation sequence ∫ f(x)f(Tnx) f(T 2nx) ... f(Tknx) dμ (x), where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d * (E) > 0 and for all ∈ ;> 0, the set {n ∈ ℤ: d* (E ∪ ( E + n) ∪ (E + 2n) ∪ (E + 3n)) > d* (E)4 - ∈} is syndetic.

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U2 - 10.1007/s00222-004-0428-6

DO - 10.1007/s00222-004-0428-6

M3 - Article

AN - SCOPUS:17444412724

SN - 0020-9910

VL - 160

SP - 261

EP - 303

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 2

ER -