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## The quantitative behaviour of polynomial orbits on nilmanifolds (2007)

Citations: | 38 - 3 self |

### Citations

417 | Recurrence in Ergodic Theory and Combinatorial Number Theory, - Furstenberg - 1981 |

209 |
Regular partitions of graphs, Problemes combinatoires et theorie des graphes
- Szemeredi
- 1976
(Show Context)
Citation Context ...) × (periodic). The notion of a subgroup G ′ being M-rational relative to a Mal’cev basis X will be defined in Definition 2.6. This result has some faint resemblance to the Szemerédi regularity lemma =-=[26]-=-, although with the key difference that our bounds here are all polynomial in nature. The derivation of Theorem 1.23 from Theorem 1.20 will be performed in §11-13. We will use Theorem 1.23 in [13] in ... |

197 |
Representation of nilpotent Lie groups and their applications. Part 1: Basic theory and examples
- Corwin
- 1990
(Show Context)
Citation Context ...H ∩ Γ is a cocompact subgroup of H). We say that H is proper if H ̸= G. Example. If G/Γ is a nilmanifold, one can show that each member Gi of the lower central series is a rational subgroup; see e.g. =-=[4]-=- or [22]. ⋄ Definition 1.14 (Rational sequence). Let G/Γ be a nilmanifold. A rational group element is any g ∈ G such that g r ∈ Γ for some integer r > 0. A rational point is any point in G/Γ of the f... |

183 |
On a class of homogeneous spaces
- Mal’cev
- 1949
(Show Context)
Citation Context ...s a cocompact subgroup of H). We say that H is proper if H ̸= G. Example. If G/Γ is a nilmanifold, one can show that each member Gi of the lower central series is a rational subgroup; see e.g. [4] or =-=[22]-=-. ⋄ Definition 1.14 (Rational sequence). Let G/Γ be a nilmanifold. A rational group element is any g ∈ G such that g r ∈ Γ for some integer r > 0. A rational point is any point in G/Γ of the form gΓ f... |

148 | Non-conventional ergodic averages and nilmanifolds,
- Host, Kra
- 2005
(Show Context)
Citation Context ... them, play a fundamental rôle in combinatorial number theory. Their relevance was certainly apparent in [8], and it has been displayed quite dramatically in recent ergodic-theoretic work of Host-Kra =-=[15]-=- and Ziegler [31]. More recently the authors have explored how nilmanifolds arise in additive combinatorics [10] and in the study of linear equations in the primes [12]. The present paper is a part of... |

105 |
Raghunathan’s topological conjecture and distributions of unipotent flows,
- Ratner
- 1991
(Show Context)
Citation Context ...e of a general, not necessarily nilpotent, Lie group) play a fundamental rôle in number theory; see [30] for a discussion. These questions are also closely related to the celebrated theorem of Ratner =-=[24]-=- on unipotent flows, although as we are restricting attention to nilmanifolds, we will not need the full force of Ratner’s theorem (or quantitative versions thereof) here. 1.4. Qualitative equidistrib... |

92 |
On triples in arithmetic progression
- Bourgain
- 1999
(Show Context)
Citation Context ...f additive combinatorics,THE QUANTITATIVE BEHAVIOUR OF POLYNOMIAL ORBITS ON NILMANIFOLDS 9 particularly those parts of it that have the flavour of “quantitative ergodic theory”. The work of Bourgain =-=[3]-=- on Roth’s theorem is another example. ⋄ Of course, by specialising to linear sequences, Theorem 1.20 also implies a quantitative version of Leon Green’s theorem. A separate proof of this linear case ... |

91 | Universal characteristic factors and Furstenberg averages,
- Ziegler
- 2007
(Show Context)
Citation Context ...damental rôle in combinatorial number theory. Their relevance was certainly apparent in [8], and it has been displayed quite dramatically in recent ergodic-theoretic work of Host-Kra [15] and Ziegler =-=[31]-=-. More recently the authors have explored how nilmanifolds arise in additive combinatorics [10] and in the study of linear equations in the primes [12]. The present paper is a part of that programme (... |

79 | Linear equations in primes.
- Green, Tao
- 2010
(Show Context)
Citation Context ...ur bounds are uniform in N and are polynomial in the error tolerance δ. In a subsequent paper [13] we shall use this theorem to establish the Möbius and Nilsequences conjecture from our earlier paper =-=[12]-=-. 1. Introduction 1.1. Nilmanifolds. In the last few years it has come to be appreciated that nilmanifolds, together with orbits on them, play a fundamental rôle in combinatorial number theory. Their ... |

54 |
Flows on homogeneous spaces,
- Auslander, Green, et al.
- 1963
(Show Context)
Citation Context ...d only if the projected orbit (π(g(n)Γ))n∈N is equidistributed in the horizontal torus G/(G2Γ). (In particular, (g(n)Γ)n∈Z is equidistributed if and only if it is totally equidistributed.) Proof. See =-=[1, 14]-=-. Leon Green used representation theory to establish his result, but a more elementary proof was subsequently found by Parry [23]. Example. Suppose that G/Γ is the Heisenberg example (1.1). Then ) G2 ... |

41 |
Sur les groupes nilpotents et les anneaux de
- Lazard
- 1954
(Show Context)
Citation Context ...efully that the degree of a linear sequence is equal to the step s of the underlying Lie group G, and is not equal to one as the name “linear” might suggest. A remarkable result of Lazard and Leibman =-=[19, 20, 21]-=- asserts that poly(Z, G•) is a group. We will prove this in §6, and it will play a key rôle in several of our arguments. Theorem 1.6 was extended by Liebman [22] to the case when g(n) is a polynomial ... |

35 |
A contribution to the theory of groups of prime power order
- Hall
- 1934
(Show Context)
Citation Context ...sequently in [12, Appendix E]. See also the recent preprint [17]. We thank Sasha Leibman for helpful conversations concerning these methods. One should mention at this point the Hall-Petresco theorem =-=[15, 27]-=-, which established a special case of the Lazard-Leibman theorem. This theorem states that if G• is the lower central series filtration then the sequence n ↦→ a n b n lies in poly(Z, G•) for any a, b ... |

28 | Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Lie groups and ergodic theory
- Shah
- 1996
(Show Context)
Citation Context ...not equidistributed, then this sequence instead takes values in a finite union of proper subtori of G/Γ. This can be viewed as an extremely simple special case of the theorems of Ratner [24] and Shah =-=[25]-=-. More quantitative results can be obtained via Fourier analysis 1 ; see Proposition 3.1 below. A remarkable theorem of Leon Green allows one to reduce qualitative questions about the distribution of ... |

25 |
Polynomial sequences in groups,
- Leibman
- 1998
(Show Context)
Citation Context ...t there is an equivalent and rather neater definition: g : Z → G is polynomial if and only if ∂ l g(n) = idG for all large integers l, where ∂g(n) := g(n + 1)g(n) −1 . The equivalence (due to Leibman =-=[17]-=-) is not entirely trivial; see Lemma 7.9. One key advantage of working with polynomial sequences instead of linear ones is that they manifestly form a group; the product of two polynomial sequences of... |

24 | Multiple recurrence and nilsequences,”
- Bergelson, Host, et al.
- 2005
(Show Context)
Citation Context ...applies to multiparameter polynomial mappings g : Z t → G. In the infinitary setting such a generalization was obtained by Leibman [20], and his result has subsequently been applied in such papers as =-=[2]-=- and [21]. We have taken the trouble to derive multiparameter extensions of our main results with analogous finitary applications in mind; see Theorems 11.8 and Theorem 13.2. 2. Precise statements of ... |

24 |
The Hardy-Littlewood Method. Cambridge Tracts in
- Vaughan
- 1981
(Show Context)
Citation Context ...NTITATIVE BEHAVIOUR OF POLYNOMIAL ORBITS ON NILMANIFOLDS 21 this is really just a reprise of the standard theory of Weyl sums as used, for example, in the study of Waring’s problem (see, for example, =-=[29]-=-). Lemma 4.1 (van der Corput inequality). Let N, H be positive integers and suppose that (an)n∈[N] is a sequence of complex numbers. Extend (an) to all of Z by defining an := 0 when n /∈ [N]. Then |En... |

23 |
Dynamical systems on nilmanifolds
- Parry
- 1970
(Show Context)
Citation Context ...stributed if and only if it is totally equidistributed.) Proof. See [1, 14]. Leon Green used representation theory to establish his result, but a more elementary proof was subsequently found by Parry =-=[23]-=-. Example. Suppose that G/Γ is the Heisenberg example (1.1). Then ) G2 = ( 1 0 R 0 1 0 0 0 1 and G/G2Γ may be identified with R2 /Z2, the projection π being given by )] π := (x1, x3). [( 1 x1 x2 0 1 x... |

19 |
Ratner’s Theorems on Unipotent Flows, Chicago Lectures in Mathematics. University of Chicago Press
- Morris
- 2005
(Show Context)
Citation Context ...tained a similar result for arbitrary discrete unipotent (but linear) flows on a finite volume homogeneous space; the case of continuous unipotent linear flows was treated earlier by Ratner [24] (see =-=[5]-=- for further discussion). Leibman’s proof of Corollary 1.16 does not use these results, but instead proceeds in two stages. Firstly, by iterating Theorem 1.8 (or more precisely a generalization of thi... |

17 | Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces,
- Einsiedler, Margulis, et al.
- 2009
(Show Context)
Citation Context ...)n∈Pi lies within δ (using the metric dX) of xiHΓ/Γ ⊆ G/Γ, where xi ∈ G and H is a closed subgroup of G which is δ −Os,m,d(1) -rational relative to X. Remark. The reader may wish to compare this with =-=[6]-=-, another recent result on quantitative variants of Ratner’s theorem. Let us conclude this introduction by remarking that our main theorem actually applies to multiparameter polynomial mappings g : Z ... |

14 | From combinatorics to ergodic theory and back again
- Kra
- 2006
(Show Context)
Citation Context ...o this paper) but, since it concerns only the intrinsic properties of nilmanifolds, may be read independently of any of the other work. The reader interested in the background may consult the surveys =-=[16, 9, 27]-=- or the paper [12]. We begin by setting out our notation for nilmanifolds. Definition 1.2 (Nilmanifolds). Let G be a connected, simply connected Lie group with identity element idG. We define the lowe... |

12 |
Nonconventional ergodic averages, The legacy of John von Neumann
- Furstenberg
- 1988
(Show Context)
Citation Context ... In the last few years it has come to be appreciated that nilmanifolds, together with orbits on them, play a fundamental rôle in combinatorial number theory. Their relevance was certainly apparent in =-=[8]-=-, and it has been displayed quite dramatically in recent ergodic-theoretic work of Host-Kra [15] and Ziegler [31]. More recently the authors have explored how nilmanifolds arise in additive combinator... |

12 | Generalising the Hardy-Littlewood method for primes,
- Green
- 2006
(Show Context)
Citation Context ...o this paper) but, since it concerns only the intrinsic properties of nilmanifolds, may be read independently of any of the other work. The reader interested in the background may consult the surveys =-=[16, 9, 27]-=- or the paper [12]. We begin by setting out our notation for nilmanifolds. Definition 1.2 (Nilmanifolds). Let G be a connected, simply connected Lie group with identity element idG. We define the lowe... |

12 |
convergence of ergodic averages of polynomial sequences of translations on a nilmanifold, Ergodic Theory and Dynamical Systems 25
- Pointwise
(Show Context)
Citation Context ...f linear ones is that they manifestly form a group; the product of two polynomial sequences of degree at most d is again a polynomial sequence of degree at most d. Theorem 1.8 was extended by Liebman =-=[19]-=- to the case when g(n) was a polynomial sequence rather than a linear one. In particular, he showed the following generalisation of Theorem 1.9: Theorem 1.12 (Leibman’s theorem). [19] Suppose that G/Γ... |

9 |
An inverse theorem for the Gowers U 3-norm, with applications
- Green, Tao
(Show Context)
Citation Context ...d it has been displayed quite dramatically in recent ergodic-theoretic work of Host-Kra [15] and Ziegler [31]. More recently the authors have explored how nilmanifolds arise in additive combinatorics =-=[10]-=- and in the study of linear equations in the primes [12]. The present paper is a part of that programme (and in particular will be used to prove the Möbius and Nilsequences conjecture from [12] in the... |

8 | Obstructions to uniformity and arithmetic patterns in the primes,
- Tao
- 2006
(Show Context)
Citation Context ...o this paper) but, since it concerns only the intrinsic properties of nilmanifolds, may be read independently of any of the other work. The reader interested in the background may consult the surveys =-=[16, 9, 27]-=- or the paper [12]. We begin by setting out our notation for nilmanifolds. Definition 1.2 (Nilmanifolds). Let G be a connected, simply connected Lie group with identity element idG. We define the lowe... |

7 |
Sur les commutateurs
- Petresco
- 1954
(Show Context)
Citation Context ...sequently in [12, Appendix E]. See also the recent preprint [17]. We thank Sasha Leibman for helpful conversations concerning these methods. One should mention at this point the Hall-Petresco theorem =-=[15, 27]-=-, which established a special case of the Lazard-Leibman theorem. This theorem states that if G• is the lower central series filtration then the sequence n ↦→ a n b n lies in poly(Z, G•) for any a, b ... |

5 |
Spectra of nilflows
- Green
- 1961
(Show Context)
Citation Context ...d only if the projected orbit (π(g(n)Γ))n∈N is equidistributed in the horizontal torus G/(G2Γ). (In particular, (g(n)Γ)n∈Z is equidistributed if and only if it is totally equidistributed.) Proof. See =-=[1, 14]-=-. Leon Green used representation theory to establish his result, but a more elementary proof was subsequently found by Parry [23]. Example. Suppose that G/Γ is the Heisenberg example (1.1). Then ) G2 ... |

5 |
convergence of ergodic averages for polynomial actions of Zd by translations on a nilmanifold, Ergodic Theory and Dynamical Systems 25
- Pointwise
(Show Context)
Citation Context ...nclude this introduction by remarking that our main theorem actually applies to multiparameter polynomial mappings g : Z t → G. In the infinitary setting such a generalization was obtained by Leibman =-=[20]-=-, and his result has subsequently been applied in such papers as [2] and [21]. We have taken the trouble to derive multiparameter extensions of our main results with analogous finitary applications in... |

5 |
On triples in arithmetic progression, Geom
- Bourgain, J
- 1999
(Show Context)
Citation Context ...n a variety of different scales like this is very much a feature of additive combinatorics, particularly those parts of it that have the flavour of “quantitative ergodic theory”. The work of Bourgain =-=[3]-=- on Roth’s theorem is another example. ⋄ Of course, by specialising to linear sequences, Theorem 1.16 also implies a quantitative version of Leon Green’s theorem. The proof of Theorem 1.16 could be si... |

4 |
mappings of groups,
- Polynomial
- 2002
(Show Context)
Citation Context ...i ji ) . 7 In all likelihood one could analyse the multiparameter situation directly using the methods of §6 – §10. Much of §7 may be generalized for polynomial mappings g : Z t → G, for example: see =-=[18]-=-. For pedagogical reasons we have chosen not to follow this path, since the deduction of the multiparameter theory from the one-parameter theory is relatively painless.THE QUANTITATIVE BEHAVIOUR OF P... |

4 |
Spectral theory of automorphic forms, a very brief introduction, in Equidistribution in Number Theory, an introduction
- Venkatesh
(Show Context)
Citation Context ...ns, and corresponding questions in more general settings (for example when G/Γ is a homogeneous space of a general, not necessarily nilpotent, Lie group) play a fundamental rôle in number theory; see =-=[30]-=- for a discussion. These questions are also closely related to the celebrated theorem of Ratner [24] on unipotent flows, although as we are restricting attention to nilmanifolds, we will not need the ... |

1 |
of the diagonal of the power of a nilmanifold, preprint
- Orbit
(Show Context)
Citation Context ...to multiparameter polynomial mappings g : Z t → G. In the infinitary setting such a generalization was obtained by Leibman [20], and his result has subsequently been applied in such papers as [2] and =-=[21]-=-. We have taken the trouble to derive multiparameter extensions of our main results with analogous finitary applications in mind; see Theorems 11.8 and Theorem 13.2. 2. Precise statements of results I... |