### Abstract

Multiple ergodic averages, such as the average of expressions like f _{1}(T^{n}x) f_{2}(T^{2n}x). .. f _{k}(T^{kn}x), were first studied in the ergodic theoretic proof of Szemerédi's Theorem on arithmetic progressions. It turns out that all constraints on such averages (in a sense that we describe) have an algebraic character, arising from identities in nilpotent groups. We discuss these averages, several generalizations, and combinatorial implications of the results.

Original language | English (US) |
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Pages | 57-76 |

Number of pages | 20 |

State | Published - Dec 1 2006 |

Event | 25th International Congress of Mathematicians, ICM 2006 - Madrid, Spain Duration: Aug 22 2006 → Aug 30 2006 |

### Other

Other | 25th International Congress of Mathematicians, ICM 2006 |
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Country | Spain |

City | Madrid |

Period | 8/22/06 → 8/30/06 |

### Fingerprint

### Keywords

- Arithmetic progressions
- Multiple ergodic theorem
- Multiple recurrence
- Nilsystems

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*From combinatorics to ergodic theory and back again*. 57-76. Paper presented at 25th International Congress of Mathematicians, ICM 2006, Madrid, Spain.

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**From combinatorics to ergodic theory and back again.** / Kra, Bryna R.

Research output: Contribution to conference › Paper

TY - CONF

T1 - From combinatorics to ergodic theory and back again

AU - Kra, Bryna R

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Multiple ergodic averages, such as the average of expressions like f 1(Tnx) f2(T2nx). .. f k(Tknx), were first studied in the ergodic theoretic proof of Szemerédi's Theorem on arithmetic progressions. It turns out that all constraints on such averages (in a sense that we describe) have an algebraic character, arising from identities in nilpotent groups. We discuss these averages, several generalizations, and combinatorial implications of the results.

AB - Multiple ergodic averages, such as the average of expressions like f 1(Tnx) f2(T2nx). .. f k(Tknx), were first studied in the ergodic theoretic proof of Szemerédi's Theorem on arithmetic progressions. It turns out that all constraints on such averages (in a sense that we describe) have an algebraic character, arising from identities in nilpotent groups. We discuss these averages, several generalizations, and combinatorial implications of the results.

KW - Arithmetic progressions

KW - Multiple ergodic theorem

KW - Multiple recurrence

KW - Nilsystems

UR - http://www.scopus.com/inward/record.url?scp=76249091021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76249091021&partnerID=8YFLogxK

M3 - Paper

AN - SCOPUS:76249091021

SP - 57

EP - 76

ER -