### Abstract

We offer a proof of the following non-conventional ergodic theorem: If Ti:^{r}ℤ^{r}(X,μ,μ) for i=1,2,⋯,d are commuting probability-preserving ^{r}-actions, (IN)N1 is a Flner sequence of subsets of ℤ^{r}, (IN)N≥1 is a base-point sequence in ℤ^{r} and f1,f2,⋯,fd∈L∞(μ) then the non-conventional ergodic averages 1/|IN| ∑n∈IN+aN ∏i=1 converge to some limit in L^{2}(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Taos proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

Original language | English (US) |
---|---|

Pages (from-to) | 321-338 |

Number of pages | 18 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*30*(2), 321-338. https://doi.org/10.1017/S014338570900011X

**On the norm convergence of non-conventional ergodic averages.** / Austin, Tim.

Research output: Contribution to journal › Article

*Ergodic Theory and Dynamical Systems*, vol. 30, no. 2, pp. 321-338. https://doi.org/10.1017/S014338570900011X

}

TY - JOUR

T1 - On the norm convergence of non-conventional ergodic averages

AU - Austin, Tim

PY - 2010/4

Y1 - 2010/4

N2 - We offer a proof of the following non-conventional ergodic theorem: If Ti:rℤr(X,μ,μ) for i=1,2,⋯,d are commuting probability-preserving r-actions, (IN)N1 is a Flner sequence of subsets of ℤr, (IN)N≥1 is a base-point sequence in ℤr and f1,f2,⋯,fd∈L∞(μ) then the non-conventional ergodic averages 1/|IN| ∑n∈IN+aN ∏i=1 converge to some limit in L2(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Taos proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

AB - We offer a proof of the following non-conventional ergodic theorem: If Ti:rℤr(X,μ,μ) for i=1,2,⋯,d are commuting probability-preserving r-actions, (IN)N1 is a Flner sequence of subsets of ℤr, (IN)N≥1 is a base-point sequence in ℤr and f1,f2,⋯,fd∈L∞(μ) then the non-conventional ergodic averages 1/|IN| ∑n∈IN+aN ∏i=1 converge to some limit in L2(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Taos proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

UR - http://www.scopus.com/inward/record.url?scp=77951254529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951254529&partnerID=8YFLogxK

U2 - 10.1017/S014338570900011X

DO - 10.1017/S014338570900011X

M3 - Article

AN - SCOPUS:77951254529

VL - 30

SP - 321

EP - 338

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 2

ER -