## Multiple recurrence and nilsequences (2004)

Citations: | 16 - 2 self |

### BibTeX

@MISC{Bergelson04multiplerecurrence,

author = {Vitaly Bergelson and Bernard Host and Bryna Kra},

title = {Multiple recurrence and nilsequences},

year = {2004}

}

### OpenURL

### Abstract

### Citations

250 |
Recurrence in Ergodic Theory and Combinatorial Number Theory
- Furstenberg
- 1981
(Show Context)
Citation Context ...on that it suffices to prove the ergodic theoretic results for ergodic systems was transmitted to us by Lesigne (personal communication); the proof we give is almost entirely contained in Furstenberg =-=[F2]-=-. Proof. We proceed as in the proof of Lemma 3.7 of [F2]. Let @ 1G e be endowed with the product topology and the shift map T 0given by , @ ; xn8 T n 1 for 1IA all n . 1 0- 1G e Define e by ; setting ... |

249 | On sets of integers containing no k elements in arithmetic progressions
- Szemerédi
(Show Context)
Citation Context ...e infimum of the sequence. Furstenberg used his Multiple Recurrence Theorem to make the beautiful connection between ergodic theory and combinatorics and prove Szemerédi’s Theorem: Theorem (Szemerédi =-=[S]-=-). A subset of integers with positive upper Banach density contains arithmetic progressions of arbitrary finite length. Using a variation of Furstenberg’s Correspondence Principle (Proposition 3.1) an... |

155 |
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
- Furstenberg
- 1977
(Show Context)
Citation Context ... showing that under the same assumptions: For every ε 6 0, J n 1KA : µ , A 3 T n A0?6 µ , A0 2 9 ε L is syndetic. More recently, Furstenberg proved a Multiple Recurrence Theorem: Theorem (Furstenberg =-=[F1]-=-). Let , X - ./- µ- T 0 be a system, let A 1M. be a set with µ , A0N6 0 and let k < 1 an integer. Then liminf NO MPQ8 ∞ 1 N 9 M 1 NO ∑ µ nR M , A T 3 n A T 3 2n S S 3 A T 3KS kn 4 0 A0T6 The liminf is... |

115 |
On a class of homogeneous spaces
- Malcev
- 1951
(Show Context)
Citation Context ...or this action. There exists a unique Borel probability measure µ on X invariant under this action, called the Haar measure of X. The fundamental properties of nilmanifolds were established by Malcev =-=[M]-=-. We make use of the following property several times, which appears in [M] for connected groups and is proved in Leibman [Lei2] in a similar way for the general case: – For every integer j < 1, the s... |

83 |
Nonconventional ergodic averages and nilmanifolds
- Host, Kra
(Show Context)
Citation Context ... be a system, let A 1M. be a set with µ , A0N6 0 and let k < 1 an integer. Then liminf NO MPQ8 ∞ 1 N 9 M 1 NO ∑ µ nR M , A T 3 n A T 3 2n S S 3 A T 3KS kn 4 0 A0T6 The liminf is actually a limit; see =-=[HK]-=-. (See also [Z2].) In particular, there exist infinitely many integers n such that µ , A3 T n A3 T 2n A 3US S S 3 T kn A0V6 0. Furstenberg’s Theorem can thus be considered as a far reaching generaliza... |

79 |
Strict ergodicity and transformation of the torus
- Furstenberg
- 1961
(Show Context)
Citation Context ... Kronecker factor is Z ; G2Λ with the rotation induced by t and with the natural factor map G_ ; G_ X } G_ Λ ; G2Λ Z. For connected groups, parts 1, 2 and 3 of this theorem can be deduced from [AGH], =-=[F3]-=- and [Pa1], while parts 4 and 6 are proved in [Pa1]. When G is connected and simply connected and, more generally, when G can be imbedded as a closed subgroup of a connected simply connected k-step ni... |

49 | Universal Characteristic Factors and FurstenbergAverages
- Ziegler
(Show Context)
Citation Context ...t A 1M. be a set with µ , A0N6 0 and let k < 1 an integer. Then liminf NO MPQ8 ∞ 1 N 9 M 1 NO ∑ µ nR M , A T 3 n A T 3 2n S S 3 A T 3KS kn 4 0 A0T6 The liminf is actually a limit; see [HK]. (See also =-=[Z2]-=-.) In particular, there exist infinitely many integers n such that µ , A3 T n A3 T 2n A 3US S S 3 T kn A0V6 0. Furstenberg’s Theorem can thus be considered as a far reaching generalization of the Poin... |

36 |
Flows on homogeneous spaces
- Auslander, Green, et al.
- 1963
(Show Context)
Citation Context ... X ±} x S t x. Then , - T0 X is called a k-step topological nilsystem and , - µ- X 0 T is called a k-step nilsystem. Fundamental properties of nilsystems were established by Auslander, Green and Hahn =-=[AGH]-=- and by Parry [Pa1]. Further ergodic properties were proven by Parry [Pa2] and Lesigne [Les] when the group G is connected, and generalized by Leibman [Lei2]. We summarize various properties of nilsys... |

32 |
On the sets of integers which contain no three in arithmetic progression
- Behrend
- 1946
(Show Context)
Citation Context ...measure on the torus nT;€zX_ A . 2.1. A counterexample for a nonergodic system In order to show that ergodicity is necessary in Theorem 1.2, we use the following result of Behrend: 6 Theorem (Behrend =-=[Beh]-=-). For every integer L 4 4 4 - 0- 1- 9 1G , logL0 9 0, there exists a subset Λ of @ L 9 having more than Lexp , c elements that does not contain any nontrivial arithmetic progression of length 3. Theo... |

30 |
Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam
- Leibman
(Show Context)
Citation Context ...f X. The fundamental properties of nilmanifolds were established by Malcev [M]. We make use of the following property several times, which appears in [M] for connected groups and is proved in Leibman =-=[Lei2]-=- in a similar way for the general case: – For every integer j < 1, the subgroups G j and ΛGj are closed in G. It follows that the group Λ j ; Λ 3 G j is cocompact in G j.16 Vitaly Bergelson et al. X ... |

22 |
Ergodic properties of affine transformations and flows on nilmanifolds
- Parry
- 1969
(Show Context)
Citation Context ... , - T0 X is called a k-step topological nilsystem and , - µ- X 0 T is called a k-step nilsystem. Fundamental properties of nilsystems were established by Auslander, Green and Hahn [AGH] and by Parry =-=[Pa1]-=-. Further ergodic properties were proven by Parry [Pa2] and Lesigne [Les] when the group G is connected, and generalized by Leibman [Lei2]. We summarize various properties of nilsystems that we need: ... |

16 | Eine Verschärfung des Poincaréschen “Wiederkehrsatzes - Khintchine - 1934 |

15 | A non-conventional ergodic theorem for a nilsystem. Ergodic Theory Dynam. Systems 25
- Ziegler
- 2005
(Show Context)
Citation Context ...mes it is convenient to use both notations in the same formula. For f L 1 ∞, , , we first study the averages of the sequence k- n0 If . µ0 This establishes a short proof of a recent result by Ziegler =-=[Z1]-=-. We use some algebraic constructions based on ideas of Petresco [Pe] and Leibman [Lei1]. We explain the idea behind this construction. It is natural to define an arithmetic progression of : length k ... |

12 |
Dynamical systems on nilmanifolds
- Parry
- 1970
(Show Context)
Citation Context ..., - µ- X 0 T is called a k-step nilsystem. Fundamental properties of nilsystems were established by Auslander, Green and Hahn [AGH] and by Parry [Pa1]. Further ergodic properties were proven by Parry =-=[Pa2]-=- and Lesigne [Les] when the group G is connected, and generalized by Leibman [Lei2]. We summarize various properties of nilsystems that we need: Theorem 4.1. Let , X ; G_ Λ - µ- T 0 be a k-step nilsys... |

10 |
Polynomial sequences in groups
- Leibman
- 1998
(Show Context)
Citation Context ...rst study the averages of the sequence k- n0 If . µ0 This establishes a short proof of a recent result by Ziegler [Z1]. We use some algebraic constructions based on ideas of Petresco [Pe] and Leibman =-=[Lei1]-=-. We explain the idea behind this construction. It is natural to define an arithmetic progression of : length k 1 in G as an element of Gk8 1 of the form , h g- hg- 2 4 4 - h g- 4 k for g- some 1 h G.... |

8 |
une nil-variété, les parties minimales associeèe á une translation sont uniquement ergodiques. Ergodic Theory Dynam. Systems 11
- Lesigne, Sur
- 1991
(Show Context)
Citation Context ...lled a k-step nilsystem. Fundamental properties of nilsystems were established by Auslander, Green and Hahn [AGH] and by Parry [Pa1]. Further ergodic properties were proven by Parry [Pa2] and Lesigne =-=[Les]-=- when the group G is connected, and generalized by Leibman [Lei2]. We summarize various properties of nilsystems that we need: Theorem 4.1. Let , X ; G_ Λ - µ- T 0 be a k-step nilsystem with T the tra... |

7 |
Eigenfunctions of T × S and the Conze-Lesigne algebra. Ergodic Theory and its Connections with Harmonic Analysis, Eds.: Petersen
- Rudolph
- 1995
(Show Context)
Citation Context .... While an inverse limit of compact abelian Lie groups is a compact group, an inverse limit of k-step nilmanifolds is not, in general, the homogeneous space of some locally compact group (see Rudolph =-=[R]-=-). This explains why the definition of a nilsequence is not a direct generalization of the definition of an almost periodic sequence. The general decomposition result is: Theorem 1.9. Let , - ./- µ- X... |

4 |
Sur les commutateurs
- Petresco
- 1954
(Show Context)
Citation Context ... L 1 ∞, , , we first study the averages of the sequence k- n0 If . µ0 This establishes a short proof of a recent result by Ziegler [Z1]. We use some algebraic constructions based on ideas of Petresco =-=[Pe]-=- and Leibman [Lei1]. We explain the idea behind this construction. It is natural to define an arithmetic progression of : length k 1 in G as an element of Gk8 1 of the form , h g- hg- 2 4 4 - h g- 4 k... |

1 |
Weakly mixing PET
- Bergelson
- 1987
(Show Context)
Citation Context ... we write UD-Liman ; 0, if lim NPQ8 ∞ sup 1 M8 NO 1 M ∑ w an nR M ; 0 4 Equivalently, ; UD-Liman 0 if and only if for any 6 ε 0, the set @ 1 n : A w an εG has upper Banach density zero (cf. Bergelson =-=[Ber]-=-, defini6 tion 3.5). The sequence @ tσ d , G is almost periodic, and hence there exists a comn0 pact abelian group G, a continuous real valued function φ on G, and 1 a G such that tσ d , φ n0x; , an f... |