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## Some open problems on multiple ergodic averages (2011)

Citations: | 2 - 1 self |

### Citations

415 |
Recurrence in ergodic theory and combinatorial number theory
- Fürstenberg
- 1981
(Show Context)
Citation Context ... (for polynomial iterates though there is some progress in this direction [11, 12]). 2.8. Furstenberg correspondence principle. We frequently use the following correspondence principle of Furstenberg =-=[91, 92]-=- (the formulation given is from [23]) in order to reformulate statements in combinatorics as multiple recurrence statements in ergodic theory: Furstenberg Correspondence Principle. Let ℓ, d ∈ N, E ⊂ Z... |

266 | The primes contain arbitrarily long arithmetic progressions
- Green, Tao
(Show Context)
Citation Context ...ive upper density contains patterns of a certain sort could be an important first step towards proving an analogous result for the set of primes. This idea originates from work of B. Green and T. Tao =-=[104]-=-, where it was used to show that the primes contain arbitrarily long arithmetic 1The upper density d̄(E) of a set E ⊂ Zd is defined by d̄(E) := lim supN→∞ |E∩[−N,N]d| |[−N,N]d| . 4 NIKOS FRANTZIKINAKI... |

256 | A new proof of Szemerédi’s theorem
- Gowers
(Show Context)
Citation Context ...tructure. Very often this step is carried out by controlling the L2(µ)-norm of the averages (3) by the Gowers-Host-Kra seminorms. Similar seminorms were first introduced in combinatorics by T. Gowers =-=[103]-=- and their ergodic variant (that is more relevant for our study) was introduced by B. Host and B. Kra [115]. For an ergodic system (X,X , µ, T ) and function f ∈ L∞(µ), they are defined as follows: ||... |

248 |
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
- Furstenberg
- 1977
(Show Context)
Citation Context ...lent places to look for such topics, for instance, the survey articles of V. Bergelson [24, 27, 28] cover a lot of related material and contain an extensive bibliography up to 2006. Linear sequences: =-=[2, 3, 5, 6, 7, 9, 10, 14, 15, 16, 17, 18, 25, 56, 33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]-=-. Polynomial sequences: [11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]. Other s... |

197 | Representation of nilpotent Lie groups and their applications. Part 1: Basic theory and examples - Corwin - 1990 |

191 |
An introduction to infinite ergodic theory
- Aaronson
- 1997
(Show Context)
Citation Context ...involving iterates given by random sequences of integers. The random sequences that we work with are constructed by selecting a positive integer n to be a member of our sequence with probability σn ∈ =-=[0, 1]-=-. More precisely, let (Ω,F ,P) be a probability space, and let (Xn)n∈N be a sequence of independent 0−1 valued random variables with P(Xn = 1) := σn and P(Xn = 0) := 1− σn where σn is a decreasing seq... |

150 | Ergodic Theory via Joinings - Glasner - 2003 |

123 |
Polynomial extensions of Van Der Waerden’s and Szemerédi’s theorems
- Bergelson, Leibman
(Show Context)
Citation Context ...ave the first proof of the multidimensional Szemerédi theorem [97] and the density Hales-Jewett theorem [100], and V. Bergelson and A. Leibman proved the polynomial extension of Szemerédi’s theorem =-=[35]-=- (currently no proof that avoids ergodic theory is known for this result). And the story does not end there, in the last two decades new powerful tools in ergodic theory were developed and used, and a... |

113 | An ergodic Szemeredi theorem for commuting transformations,
- Furstenberg, Katznelson
- 1978
(Show Context)
Citation Context ...tenberg in [91] to give an alternate proof of Szemerédi’s theorem using ergodic theory. Subsequently, H. Furstenberg and Y. Katznelson gave the first proof of the multidimensional Szemerédi theorem =-=[97]-=- and the density Hales-Jewett theorem [100], and V. Bergelson and A. Leibman proved the polynomial extension of Szemerédi’s theorem [35] (currently no proof that avoids ergodic theory is known for th... |

102 | Tauberian theorems - Wiener - 1932 |

82 | A density version of the Hales–Jewett theorem.
- Furstenberg, Katznelson
- 1991
(Show Context)
Citation Context ... of Szemerédi’s theorem using ergodic theory. Subsequently, H. Furstenberg and Y. Katznelson gave the first proof of the multidimensional Szemerédi theorem [97] and the density Hales-Jewett theorem =-=[100]-=-, and V. Bergelson and A. Leibman proved the polynomial extension of Szemerédi’s theorem [35] (currently no proof that avoids ergodic theory is known for this result). And the story does not end ther... |

79 |
Pointwise ergodic theorems for arithmetic sets,”
- Bourgain
- 1989
(Show Context)
Citation Context ...33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]. Polynomial sequences: =-=[11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]-=-. Other sequences (smooth, random, prime numbers, generalized polynomials): [30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]. Equidistribution on ... |

56 | Ergodic Theory (with a view towards Number Theory - Einsiedler, Ward - 2011 |

52 |
On the maximal ergodic theorem for certain subsets of the integers
- Bourgain
- 1988
(Show Context)
Citation Context ...Show that the averages 1 N N∑ n=1 f1(T nx) · f2(T 2nx) · f3(T 3nx), or the averages 1 N N∑ n=1 f1(T nx) · f2(T n2x), converge pointwise. Pointwise convergence of the averages (11) is known when ℓ = 1 =-=[52]-=- and is also known when ℓ = 2 and both polynomials are linear [55] (see also [68] for an alternate proof). In all other cases the problem is open even for weak mixing systems. Partial results that dea... |

48 | On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam.
- Austin
- 2010
(Show Context)
Citation Context ...the precise statement here because this would require to introduce too much additional notation. 2.7. Pleasant and magic extensions. Motivated by the work of T. Tao [181], H. Towsner [186], T. Austin =-=[9]-=-, and B. Host [111], introduced new tools that help us handle some multiple ergodic averages. In particular, a key conceptual breakthrough that first appeared in [9], is that in some instances by work... |

45 | The ergodic theoretical proof of Szemeredi’s theorem,
- Furstenberg, Katznelson, et al.
- 1982
(Show Context)
Citation Context ...lent places to look for such topics, for instance, the survey articles of V. Bergelson [24, 27, 28] cover a lot of related material and contain an extensive bibliography up to 2006. Linear sequences: =-=[2, 3, 5, 6, 7, 9, 10, 14, 15, 16, 17, 18, 25, 56, 33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]-=-. Polynomial sequences: [11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]. Other s... |

42 |
Theoremes ergodiques pour des mesures diagonales,
- Conze, Lesigne
- 1984
(Show Context)
Citation Context ...iously one has dWmin(P) ≤ dmin(P). Some bounds in the other direction are given in [146]. Mean convergence of the averages (11) was established after a long series of intermediate results; the papers =-=[91, 63, 64, 65, 101, 173, 112, 115, 196]-=- dealt with the important case of linear polynomials, and using the machinery introduced in [115], convergence for arbitrary polynomials was finally obtained by B. Host and B. Kra in [116] except for ... |

41 |
Double recurrence and almost sure convergence
- Bourgain
- 1990
(Show Context)
Citation Context ..., or the averages 1 N N∑ n=1 f1(T nx) · f2(T n2x), converge pointwise. Pointwise convergence of the averages (11) is known when ℓ = 1 [52] and is also known when ℓ = 2 and both polynomials are linear =-=[55]-=- (see also [68] for an alternate proof). In all other cases the problem is open even for weak mixing systems. Partial results that deal with special classes of transformations can be found in [3, 5, 1... |

40 |
Weakly mixing PET, Ergodic Theory Dynam.
- Bergelson
- 1987
(Show Context)
Citation Context ...stion technique. We explain a technique that is often used to produce seminorm estimates of the type (5). It is based on an induction scheme (often called PET induction) introduced by V. Bergelson in =-=[22]-=-. Let F := {a1, . . . , aℓ} be a family of real valued sequences, and suppose that one wishes to establish seminorm estimates of the form (6) lim sup N→∞ ‖AN (f1, . . . , fℓ)‖L2(µ) ≪ min i=1,...,ℓ |||... |

39 | B.: Ergodic theory and configurations in sets of positive density
- Furstenberg, Katznelson, et al.
- 1990
(Show Context)
Citation Context ...lent places to look for such topics, for instance, the survey articles of V. Bergelson [24, 27, 28] cover a lot of related material and contain an extensive bibliography up to 2006. Linear sequences: =-=[2, 3, 5, 6, 7, 9, 10, 14, 15, 16, 17, 18, 25, 56, 33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]-=-. Polynomial sequences: [11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]. Other s... |

35 |
Sur un théoréme ergodique pour des mesures diagonales
- Conze, Lesigne
- 1988
(Show Context)
Citation Context ...ind several examples demonstrating this approach, or variants of it, to prove multiple recurrence and convergence results, as well as related applications in combinatorics, in the following articles: =-=[7, 33, 40, 41, 58, 61, 62, 64, 76, 77, 79, 82, 83, 84, 85, 87, 90, 101, 108, 114, 115, 116, 119, 120, 124, 142, 154, 168, 173, 191, 195, 196]-=-. Depending on the problem, the difficulty of each step varies; typically the first step is elementary and is carried out by successive uses of the Cauchy-Schwarz inequality and an estimate of van der... |

30 |
Ergodic Ramsey Theory – an Update, Ergodic Theory
- Bergelson
- 1996
(Show Context)
Citation Context ...he richness of the return times in various multiple recurrence results). For material and a list of problems that goes beyond the scope of this set of notes we refer the reader to the survey articles =-=[24, 27, 28]-=- and the references therein. Whenever appropriate, I include the “original source” and a short history of the problem, as well as a hopefully accurate and up to date list of related work already done.... |

30 | Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam.
- Frantzikinakis, Kra
- 2005
(Show Context)
Citation Context ...A ∩ T−p1(n)1 A ∩ · · · ∩ T−pℓ(n)ℓ A) ≥ µ(A)ℓ+1 − ε. In fact, the set of integers n for which (13) holds is expected to have bounded gaps. The lower bounds are known when all transformations are equal =-=[84]-=- and they are also known for general commuting transformations when the polynomials are monomials with distinct degrees [62]. The result fails if the polynomials are distinct and pairwise dependent; i... |

27 | Distribution of values of bounded generalized polynomials,
- Bergelson, Leibman
- 2007
(Show Context)
Citation Context ..., 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]. Other sequences (smooth, random, prime numbers, generalized polynomials): =-=[30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]-=-. Equidistribution on nilmanifolds and other nil-stuff: [8, 57, 78, 106, 107, 117, 118, 120, 121, 122, 138, 140, 141, 143, 144, 145, 146, 147, 148, 152, 153, 157, 163, 164, 165, 166, 174, 175, 194]. S... |

26 | The multifarious Poincare recurrence theorem, Descriptive set theory and dynamical systems, 31–57, - Bergelson - 2000 |

23 | Pointwise convergence of ergodic averages along cubes, preprint
- Assani
(Show Context)
Citation Context ...ind several examples demonstrating this approach, or variants of it, to prove multiple recurrence and convergence results, as well as related applications in combinatorics, in the following articles: =-=[7, 33, 40, 41, 58, 61, 62, 64, 76, 77, 79, 82, 83, 84, 85, 87, 90, 101, 108, 114, 115, 116, 119, 120, 124, 142, 154, 168, 173, 191, 195, 196]-=-. Depending on the problem, the difficulty of each step varies; typically the first step is elementary and is carried out by successive uses of the Cauchy-Schwarz inequality and an estimate of van der... |

23 |
Ergodic averaging sequences
- Boshernitzan, Kolesnik, et al.
(Show Context)
Citation Context ...oblems from [79] related to the mean convergence of multiple ergodic averages involving iterates given by Hardy sequences. The following result was proved in [79] (the case ℓ = 1 was first handled in =-=[50]-=-): Theorem. Let a ∈ H have polynomial growth. Then the sequence ([a(n)]) is good for multiple convergence of powers if and only if one of the following conditions is satisfied: • |a(t)− cp(t)|/ log t→... |

23 | Poincaré recurrence and number theory - Furstenberg - 1981 |

22 |
Combinatorial and Diophantine Applications of Ergodic Theory” Handbook of dynamical systems. Vol. 1B
- Bergelson
- 2006
(Show Context)
Citation Context ...he richness of the return times in various multiple recurrence results). For material and a list of problems that goes beyond the scope of this set of notes we refer the reader to the survey articles =-=[24, 27, 28]-=- and the references therein. Whenever appropriate, I include the “original source” and a short history of the problem, as well as a hopefully accurate and up to date list of related work already done.... |

22 | Intersective polynomials and the polynomial Szemeredi theorem,
- Bergelson, Leibman, et al.
- 2008
(Show Context)
Citation Context ...ind several examples demonstrating this approach, or variants of it, to prove multiple recurrence and convergence results, as well as related applications in combinatorics, in the following articles: =-=[7, 33, 40, 41, 58, 61, 62, 64, 76, 77, 79, 82, 83, 84, 85, 87, 90, 101, 108, 114, 115, 116, 119, 120, 124, 142, 154, 168, 173, 191, 195, 196]-=-. Depending on the problem, the difficulty of each step varies; typically the first step is elementary and is carried out by successive uses of the Cauchy-Schwarz inequality and an estimate of van der... |

21 |
with an Appendix by I. Ruzsa. Multiple recurrence and nilsequences
- Bergelson, Host, et al.
(Show Context)
Citation Context |

20 | Polynomial averages converge to the product of integrals,
- Frantzikinakis, Kra
- 2005
(Show Context)
Citation Context |

19 | Multiple ergodic averages for three polynomials and applications,
- Frantzikinakis
- 2008
(Show Context)
Citation Context |

18 |
Flows on homogeneous spaces. With the assistance of
- Auslander, Green, et al.
- 1963
(Show Context)
Citation Context ... handling such equidistribution problems have been developed in recent years, thus making such a reduction very much worthwhile. Some examples of equidistribution results of this type can be found in =-=[8, 78, 106, 107, 140, 141, 152, 153]-=-. 2.4. A general strategy. Summarizing, when one is against a multiple recurrence problem in ergodic theory, or more generally any problem that can be solved by analyzing the limiting 3A measure prese... |

18 | On the two-dimensional bilinear Hilbert transform
- Demeter, Thiele
(Show Context)
Citation Context |

16 | Pleasant extensions retaining algebraic structure, II. Preprint, available online at arXiv.org: 0910.0907
- Austin
(Show Context)
Citation Context ...n about the limiting function, and also, up to now, it has not proved to be as useful when some of the iterates are non-linear (for polynomial iterates though there is some progress in this direction =-=[11, 12]-=-). 2.8. Furstenberg correspondence principle. We frequently use the following correspondence principle of Furstenberg [91, 92] (the formulation given is from [23]) in order to reformulate statements i... |

16 | A nilpotent Roth theorem, - Bergelson, Leibman - 2002 |

15 | Deducing the multidimensional Szemerédi Theorem from an infinitary removal lemma - Austin |

14 | Ergodic averages for independent polynomials and applications
- Frantzikinakis, Kra
(Show Context)
Citation Context |

13 |
Multiple recurrence of Markov shifts and other infinite measure preserving transformations
- Aaronson, Nakada
(Show Context)
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13 | IP-systems, generalized polynomials and recurrence, Ergodic Theory Dynam. - Bergelson, Haland-Knutson, et al. - 2006 |

12 |
Some results on nonlinear recurrence
- Bergelson, Boshernitzan, et al.
- 1994
(Show Context)
Citation Context ...33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]. Polynomial sequences: =-=[11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]-=-. Other sequences (smooth, random, prime numbers, generalized polynomials): [30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]. Equidistribution on ... |

12 |
Uniform distribution and Hardy fields
- Boshernitzan
- 1994
(Show Context)
Citation Context ...t)→∞. 26 NIKOS FRANTZIKINAKIS condition encodes a lot of useful information and this enables us to give more transparent and appetizing statements. Background material on Hardy fields can be found in =-=[46, 47, 48, 109, 110, 171]-=-. 6.1. Powers of a single transformation. 6.1.1. Hardy sequences of polynomial growth. To avoid repetition we remark that in this subsection we always work with a family F := {a1(t), . . . , aℓ(t)} of... |

12 | Pointwise convergence of the ergodic bilinear Hilbert transform, preprint available at http://arxiv.org/abs/math.CA/0601277
- Demeter
(Show Context)
Citation Context ...es 1 N N∑ n=1 f1(T nx) · f2(T n2x), converge pointwise. Pointwise convergence of the averages (11) is known when ℓ = 1 [52] and is also known when ℓ = 2 and both polynomials are linear [55] (see also =-=[68]-=- for an alternate proof). In all other cases the problem is open even for weak mixing systems. Partial results that deal with special classes of transformations can be found in [3, 5, 15, 16, 70, 140,... |

12 | Nonconventional ergodic averages, The legacy of John von Neumann - Furstenberg - 1988 |

11 |
Ergodic Ramsey Theory. Logic and combinatorics
- Bergelson
- 1987
(Show Context)
Citation Context ...s some progress in this direction [11, 12]). 2.8. Furstenberg correspondence principle. We frequently use the following correspondence principle of Furstenberg [91, 92] (the formulation given is from =-=[23]-=-) in order to reformulate statements in combinatorics as multiple recurrence statements in ergodic theory: Furstenberg Correspondence Principle. Let ℓ, d ∈ N, E ⊂ Zd be a set of integers, and v1, . . ... |

11 |
H̊aland, Sets of recurrence and generalized polynomials, Convergence in ergodic theory and probability
- Bergelson, J
- 1996
(Show Context)
Citation Context ...33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]. Polynomial sequences: =-=[11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]-=-. Other sequences (smooth, random, prime numbers, generalized polynomials): [30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]. Equidistribution on ... |

11 | On the set of common differences in van der Waerden’s theorem on arithmetic progressions,
- Brown, Graham, et al.
- 1999
(Show Context)
Citation Context |

11 | A Hardy field extension of Szemerédi’s theorem
- Frantzikinakis, Wierdl
(Show Context)
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10 |
Aspects of uniformity in recurrence
- Bergelson, Host, et al.
(Show Context)
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10 |
Uniformity in polynomial Szemerédi theorem, Ergodic theory of Zd-actions (Warwick 1993-1994
- Bergelson, McCutcheon
- 1996
(Show Context)
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10 | An extension of Hardy’s class L of “orders of infinity - Boshernitzan - 1981 |

10 |
New “Orders of Infinity”.
- Boshernitzan
- 1982
(Show Context)
Citation Context ...t)→∞. 26 NIKOS FRANTZIKINAKIS condition encodes a lot of useful information and this enables us to give more transparent and appetizing statements. Background material on Hardy fields can be found in =-=[46, 47, 48, 109, 110, 171]-=-. 6.1. Powers of a single transformation. 6.1.1. Hardy sequences of polynomial growth. To avoid repetition we remark that in this subsection we always work with a family F := {a1(t), . . . , aℓ(t)} of... |

10 | Multiple recurrence for two commuting transformations. To appear Ergodic Theory Dynam. Systems Available at arXiv:0912.3381
- Chu
(Show Context)
Citation Context ...eorem. This approach has proved particularly useful for handling convergence problems of multiple ergodic averages of commuting transformations with linear iterates that previously seemed intractable =-=[9, 111, 10, 59, 60]-=- (see also [13] for an application to continuous time averages). A drawback is that it does not give much information about the limiting function, and also, up to now, it has not proved to be as usefu... |

10 | Multiple recurrence and convergence for Hardy sequences of polynomial growth
- Frantzikinakis
(Show Context)
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9 |
Multiple recurrence and almost sure convergence for weakly mixing dynamical systems,
- Assani
- 1998
(Show Context)
Citation Context ...ear [55] (see also [68] for an alternate proof). In all other cases the problem is open even for weak mixing systems. Partial results that deal with special classes of transformations can be found in =-=[3, 5, 15, 16, 70, 140, 154, 155]-=-. 5.2. Commuting transformations. Throughout this section (X,X , µ) is a probability space, T1, . . . , Tℓ : X → X are commuting, invertible measure preserving transformations, f1, . . . , fℓ ∈ L∞(µ) ... |

9 | Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
- Austin, Eisner, et al.
(Show Context)
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9 | Ergodic theory and Diophantine problems. Topics in symbolic dynamics and applications - Bergelson - 2000 |

9 |
Nilspaces, nilmanifolds and their morphisms, preprint
- Camarena, Szegedy
(Show Context)
Citation Context ... prime numbers, generalized polynomials): [30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]. Equidistribution on nilmanifolds and other nil-stuff: =-=[8, 57, 78, 106, 107, 117, 118, 120, 121, 122, 138, 140, 141, 143, 144, 145, 146, 147, 148, 152, 153, 157, 163, 164, 165, 166, 174, 175, 194]-=-. SOME OPEN PROBLEMS ON MULTIPLE ERGODIC AVERAGES 33 Books and survey articles on related topics: [1, 4, 8, 23, 24, 26, 27, 28, 66, 67, 71, 89, 92, 93, 94, 95, 96, 102, 110, 126, 130, 131, 132, 133, 1... |

9 | Ergodic averages of commuting transformations with distinct degree polynomial iterates
- Chu, Franzikinakis, et al.
(Show Context)
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8 |
Failure of Roth theorem for solvable groups of exponential growth. Ergodic Theory Dynam
- Bergelson, Leibman
(Show Context)
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8 | Convergence of weighted polynomial multiple ergodic averages,”
- Chu
- 2009
(Show Context)
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7 |
Joint ergodicity and mixing
- Berend
- 1985
(Show Context)
Citation Context ...ear [55] (see also [68] for an alternate proof). In all other cases the problem is open even for weak mixing systems. Partial results that deal with special classes of transformations can be found in =-=[3, 5, 15, 16, 70, 140, 154, 155]-=-. 5.2. Commuting transformations. Throughout this section (X,X , µ) is a probability space, T1, . . . , Tℓ : X → X are commuting, invertible measure preserving transformations, f1, . . . , fℓ ∈ L∞(µ) ... |

7 | Idempotent ultrafilters, multiple weak mixing and Szemerédi’s theorem for generalized polynomials
- Bergelson, McCutcheon
(Show Context)
Citation Context ...es ([191] for powers and [42] in general), integer polynomials with zero constant term evaluated at the shifted primes ([191] for powers and [81] in general), several generalized polynomial sequences =-=[44]-=-, and some random sequences of zero density [88]. The sequence ([nc]), where c ∈ R is positive, is known to be good for multiple recurrence of powers [90] (see also [79]), but if c ∈ R \Q is greater t... |

7 |
Un théorème ergodique polynomial ponctuel pour les endomorphismes exacts et les K-systèmes
- Derrien, Lesigne
- 1996
(Show Context)
Citation Context ...ear [55] (see also [68] for an alternate proof). In all other cases the problem is open even for weak mixing systems. Partial results that deal with special classes of transformations can be found in =-=[3, 5, 15, 16, 70, 140, 154, 155]-=-. 5.2. Commuting transformations. Throughout this section (X,X , µ) is a probability space, T1, . . . , Tℓ : X → X are commuting, invertible measure preserving transformations, f1, . . . , fℓ ∈ L∞(µ) ... |

6 |
Homogeneosly distributed sequences and Poincaré sequences of integers of sublacunary growth
- Boshernitzan
- 1983
(Show Context)
Citation Context ...t)→∞. 26 NIKOS FRANTZIKINAKIS condition encodes a lot of useful information and this enables us to give more transparent and appetizing statements. Background material on Hardy fields can be found in =-=[46, 47, 48, 109, 110, 171]-=-. 6.1. Powers of a single transformation. 6.1.1. Hardy sequences of polynomial growth. To avoid repetition we remark that in this subsection we always work with a family F := {a1(t), . . . , aℓ(t)} of... |

6 |
A mean ergodic theorem for 1/N ∑N n=1 f(Tnx)g(Tn x), Convergence in ergodic theory and probability
- Furstenberg, Weiss
- 1993
(Show Context)
Citation Context ...c factor(s). Implicit use of this notion was already made on the foundational article of H. Furstenberg [91], but the term “characteristic factor” was coined in a paper of H. Furstenberg and B. Weiss =-=[101]-=-. Given a probability space (X,X , µ) and a collection of measure preserving transformations T1, . . . , Tℓ : X → X, we say that the sub-σ-algebras X1, . . . ,Xℓ of X are characteristic factors for th... |

5 | Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory,
- Bergelson, Leibman, et al.
- 2007
(Show Context)
Citation Context ...ted can be found in [77, 146]. Furthermore, in [146] a (rather complicated) algorithm is given for computing this value. Despite such progress, the following is still open (the problem is implicit in =-=[39]-=- and was stated explicitly in [146]): Problem 8. If |P| ≥ 2, show that dmin(P) ≤ |P| − 1. The estimate is known when |P| = 2, 3 ([77]) and it is open for |P| = 4. The problem is even open when one is ... |

5 |
Ergodic theorems along sequences and Hardy
- Boshernitzan, Wierdl
- 1996
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Citation Context ..., 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]. Other sequences (smooth, random, prime numbers, generalized polynomials): =-=[30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]-=-. Equidistribution on nilmanifolds and other nil-stuff: [8, 57, 78, 106, 107, 117, 118, 120, 121, 122, 138, 140, 141, 143, 144, 145, 146, 147, 148, 152, 153, 157, 163, 164, 165, 166, 174, 175, 194]. S... |

5 |
An approach to pointwise ergodic theorems
- Bourgain
- 1988
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5 | Convergence of multiple ergodic averages along cubes for several commuting transformations. Preprint. Available at http://arxiv.org/abs/0811.3953
- Chu
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Citation Context ...eorem. This approach has proved particularly useful for handling convergence problems of multiple ergodic averages of commuting transformations with linear iterates that previously seemed intractable =-=[9, 111, 10, 59, 60]-=- (see also [13] for an application to continuous time averages). A drawback is that it does not give much information about the limiting function, and also, up to now, it has not proved to be as usefu... |

5 | The structure of strongly stationary systems
- Frantzikinakis
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5 | Equidistribution of sparse sequences on nilmanifolds
- Frantzikinakis
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Citation Context ...ut for a smaller range of eligible exponents a that depends on ℓ. Under the assumption that nσn →∞, pointwise convergence is known when the transformations are given by powers of the same nilrotation =-=[78]-=- and the functions are continuous. On the other hand, when σn = n −a for some a > 1/2, ℓ = 2, and T2 = T 2 1 , mean convergence is not even known when we restrict ourselves to the class of weak mixing... |

4 |
Averages along cubes for not necessarily commuting m.p.t. Ergodic theory and related
- Assani
- 2007
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Weak mixing implies mixing of higher orders along tempered functions. Ergodic Theory Dynam
- Bergelson, H̊aland-Knutson
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Citation Context ...orm estimates of the form (6) for linear sequences [115], polynomial sequences [116, 142], block polynomials of fixed degree [90], and some sequences coming from smooth functions of polynomial growth =-=[31, 79]-=-. Notice a common desirable feature that these sequences share: after taking successive differences (meaning iterating the operation a(n) 7→ a(n+ r)− a(n))) a finite number of times we arrive to seque... |

4 | From discrete- to continuous-time ergodic theorems
- Bergelson, Leibman, et al.
- 2010
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4 |
Pointwise convergence for cubic and polynomial ergodic averages of noncommuting transformations
- Chu, Franzikinakis
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4 |
Multiple recurrence and convergence for sets related to the primes
- Frantzikinakis, Host, et al.
(Show Context)
Citation Context ..., 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82, 83, 85, 87, 101, 108, 116, 124, 137, 142, 150, 156, 158, 168, 185]. Other sequences (smooth, random, prime numbers, generalized polynomials): =-=[30, 31, 38, 42, 47, 50, 51, 72, 79, 80, 81, 88, 90, 104, 105, 125, 136, 148, 160, 161, 183, 189, 191]-=-. Equidistribution on nilmanifolds and other nil-stuff: [8, 57, 78, 106, 107, 117, 118, 120, 121, 122, 138, 140, 141, 143, 144, 145, 146, 147, 148, 152, 153, 157, 163, 164, 165, 166, 174, 175, 194]. S... |

4 | The polynomial multidimensional Szemerédi Theorem along shifted primes
- Frantzikinakis, Host, et al.
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Citation Context ... polynomials with zero constant term [35], the shifted primes ([191] for powers and [42] in general), integer polynomials with zero constant term evaluated at the shifted primes ([191] for powers and =-=[81]-=- in general), several generalized polynomial sequences [44], and some random sequences of zero density [88]. The sequence ([nc]), where c ∈ R is positive, is known to be good for multiple recurrence o... |

4 | Sets of k-recurrence but not (k+1)-recurrence - Frantzikinakis, Lesigne, et al. |

4 | Random sequences and pointwise convergence of multiple ergodic averages
- Frantzikinakis, Lesigne, et al.
(Show Context)
Citation Context ...er polynomials with zero constant term evaluated at the shifted primes ([191] for powers and [81] in general), several generalized polynomial sequences [44], and some random sequences of zero density =-=[88]-=-. The sequence ([nc]), where c ∈ R is positive, is known to be good for multiple recurrence of powers [90] (see also [79]), but if c ∈ R \Q is greater than 1, it is not known whether it is good for mu... |

4 | Recurrent ergodic structures and Ramsey theory - Furstenberg - 1990 |

4 | A polynomial Szemerédi theorem, in Combinatorics, Paul Erdős is Eight - Furstenberg - 1996 |

3 |
Multiple ergodic theorems
- Berend
- 1988
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3 |
Jointly ergodic measure-preserving transformations
- Berend, Bergelson
- 1984
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3 | Ergodic Ramsey Theory: a dynamical approach to static theorems
- Bergelson
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Citation Context ...he richness of the return times in various multiple recurrence results). For material and a list of problems that goes beyond the scope of this set of notes we refer the reader to the survey articles =-=[24, 27, 28]-=- and the references therein. Whenever appropriate, I include the “original source” and a short history of the problem, as well as a hopefully accurate and up to date list of related work already done.... |

3 | Powers of sequences and recurrence
- Frantzikinakis, Lesigne, et al.
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2 |
Pointwise convergence of nonconventional averages
- Assani
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2 | Norm convergence of continuous-time polynomial multiple ergodic averages
- Austin
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Citation Context ... particularly useful for handling convergence problems of multiple ergodic averages of commuting transformations with linear iterates that previously seemed intractable [9, 111, 10, 59, 60] (see also =-=[13]-=- for an application to continuous time averages). A drawback is that it does not give much information about the limiting function, and also, up to now, it has not proved to be as useful when some of ... |

2 |
Characterization of joint ergodicity for noncommuting transformations
- Berend, Bergelson
- 1986
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2 |
Distribution modulo 1 of some oscillating sequences III
- Berend, Boshernitzan, et al.
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Citation Context ...ore; the averages 1N ∑N n=1 T a(n)f1 ·T 2a(n)f2 · . . . ·T ℓa(n)fℓ have the same limiting value when a(n) = [n sinn] and a(n) = n. This is known for ℓ = 1, it follows from equidistribution results in =-=[19]-=- (see also related results in [20, 21]). As far as I know the problem has not been studied when ℓ ≥ 2, even for particular classes of measure preserving systems, like nilsystems or weakly mixing syste... |

2 |
The shifted primes and the multidimensional Szemerédi and polynomial van der Waerden Theorems
- Bergelson, Leibman, et al.
(Show Context)
Citation Context ...]. Further examples of sequences that are good for multiple recurrence of commuting transformations include: integer polynomials with zero constant term [35], the shifted primes ([191] for powers and =-=[42]-=- in general), integer polynomials with zero constant term evaluated at the shifted primes ([191] for powers and [81] in general), several generalized polynomial sequences [44], and some random sequenc... |

2 | On two recurrence problems
- Boshernitzan, Glasner
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Citation Context ... tori (see [92] Section 9.1, or [86]). A well known question of Y. Katznelson asks whether a set of Bohr recurrence is necessarily a set of topological recurrence (for background on this question see =-=[49, 129, 188]-=-). Although there exist examples of sets of topological recurrence that are not sets of 1-recurrence [134], all known examples are rather complicated. All these lead one to believe that a possible exa... |

2 | Ergodic Theory, Translated from the Russian by A - Cornfeld, Fomin, et al. - 1982 |

2 |
Recurrence properties of sequences of integers
- Fan, Schneider
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2 | Solvability of linear equations within weak mixing sets
- Fish
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2 | Polynomial largeness of sumsets and totally ergodic sets, see http://arxiv.org/abs/0711.3201
- Fish
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1 |
Two techniques in multiple recurrence. Ergodic theory and its connections with harmonic analysis
- Forrest
- 1993
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1 | Powers of sequences and convergence. Ergodic Theory Dynam - Frantzikinakis, Johnson, et al. |

1 | Ergodic Theory: Recurrence. Survey. Encyclopedia of complexity and Systems Science - Frantzikinakis, McCutcheon - 2009 |