Multiple recurrence and nilsequences

By Vitaly Bergelson, B. Host, Bryna Kra, Imre Z. Ruzsa. 2005

205 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1dfba85dacdaaa5f74e7d6194ec9cb8850e05459

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Nil Bohr-sets and almost automorphy of higher order (2014, background): “Bergelson-Host-Kra’ Theorem and the proof of Theorem A(2) In this section we explain how Bergelson-Host-Kra’s result in [5] is related to Problem B-II.”
An inverse theorem for the Gowers U^3 norm (2005, methodology): “o N/2 with Z/NZ in the usual manner. We now give three key examples of nilﬂows and nilsequences. 15A general (k −2)-step nilsequence is deﬁned as the uniform limit of basic (k −2)-step sequences; see [3] for further analysis of these nilsequences. AN INVERSE THEOREM FOR THE GOWERS U3(G) NORM 57 Exam”
An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications (2010, methodology): “A counterexample example of Ruzsa in the appendix to [4] shows that Theorem 1.”
Linear equations in primes (2006, background): “This leads to a slight conflict between the nomenclature of the present paper and that of (for example) [5].”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, background): “The original definition of a nilsequence in [1] assumed F was merely continuous rather than smooth, and used linear sequences n 7→ gg0 rather than polynomial sequences.”
Infinite-step nilsystems, independence and complexity (2011, background): “Moreover, assume that the group G can be spanned by the connected component of the identity and the element t (it is always possible to reduce to this case, see [3]).”
An Inverse Theorem for the Uniformity Seminorms Associated with the Action of $${{\mathbb {F}^{\infty}_{p}}}$$ (2009, background): “∩ T A) (as in [28]), and to be able to show that c(g) can be approximated by a function of polynomials in g, in the spirit of the results in [3].”
Multiple recurrence for two commuting transformations (2009, methodology): “We remark that, by the same counterexample as in [4], the hypothesis of ergodicity is necessary for this theorem to hold.”
Multiple ergodic averages for three polynomials and applications (2006, unknown): “The same result was established in [6] in the special case of the polynomial families { n, 2 n } and { n, 2 n, 3 n } .”
The quantitative behaviour of polynomial orbits on nilmanifolds (2007, methodology): “applied in such papers as [2] and [24].”
A Szemerédi-type regularity lemma in abelian groups, with applications (2003, background): “A somewhat different application of our regularity lemma is to a question raised by Bergelson, Host and Kra in [ 4 ].”
An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement) (2010, background): “The work of Furstenberg and Weiss draws attention to the role of homogeneous spaces G/Γ of nilpotent Lie groups, and subsequent work of Host and Kra [24] provides a link, in an ergodic-theoretic context, between these spaces and certain seminorms with a formal similarity to the Gowers norms under di”