Multiple recurrence and nilsequences

  title={Multiple recurrence and nilsequences},
  author={Vitaly Bergelson and Bernard Host and Bryna Kra and Imre Z. Ruzsa},
  journal={Inventiones mathematicae},
Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system $(X,\mathcal{X},\mu,T)$ and a set $A\in\mathcal{X}$ of positive measure, the set of integers n such that $\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for… 
A decomposition of multicorrelation sequences for commuting transformations along primes
We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any
Some consequences of mild ergodicity assumptions
We obtain new results on multicorrelation sequences. In particular, we prove that given a measure preserving system $(X,\mathcal{B},\mu,T_1,\dots,T_d)$ with commuting, ergodic transformations $T_i$
A good universal weight for nonconventional ergodic averages in norm
We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system $(X,{\mathcal{F}},\unicode[STIX]{x1D707},T)$
Optimal lower bounds for multiple recurrence
Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$ . We study the largeness of sets of the form
Host–Kra theory for -systems and multiple recurrence
  • O. Shalom
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2021
Let $\mathcal {P}$ be an (unbounded) countable multiset of primes (that is, every prime may appear multiple times) and let $G=\bigoplus _{p\in \mathcal {P}}\mathbb {F}_p$ . We
Extension of Wiener-Wintner double recurrence theorem to polynomials
We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case of a polynomial exponent. We show that there exists a unique set of full measure for which the
Pointwise convergence of multiple ergodic averages and strictly ergodic models
It is proved that for an ergodic system $(X,\mathcal{X},\mu, T)$, d, f_1, \ldots, f-d \in L^{\infty}(\mu)$, the averages converge $\mu$ a.e.
A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems
We investigate how spectral properties of a measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ are reflected in the multiple ergodic averages arising from that system. For certain
Topological characteristic factors and nilsystems
We prove that the maximal infinite step pro-nilfactor $X_\infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost
Nilsequences, null-sequences, and multiple correlation sequences
  • A. Leibman
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2013
Abstract A ($d$-parameter) basic nilsequence is a sequence of the form $\psi (n)= f({a}^{n} x)$,$n\in { \mathbb{Z} }^{d} $, where $x$ is a point of a compact nilmanifold $X$, $a$ is a translation on


Universal characteristic factors and Furstenberg averages
Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and
On sets of integers containing k elements in arithmetic progression
In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic
On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.
  • F. Behrend
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1946
By a modification of Salem and Spencer' method, the better estimate 1-_2/2log2 + e v(N) > N VloggN is shown.
Strict Ergodicity and Transformation of the Torus
Introduction. If T is a measure preserving transformation ofl a probability space Q with measure Iu, the ergodic theorem assures the existence N-1 almost everywhere with respect to /i of the average
Polynomial Sequences in Groups
Abstract Given a groupGwith lower central seriesG = G1 ⊇ G2 ⊇ G3 ⊇ ···, we say that a sequenceg: Z  → Gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference
Nonconventional ergodic averages and nilmanifolds
We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions
Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques
  • E. Lesigne
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1991
Abstract We call nilmanifold every compact space X on which a connected locally compact nilpotent group acts transitively. We show that, if X is a nilmanifold and f is a continuous function on X,
Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold
  • A. Leibman
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2004
We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This
Eine Verschärfung des Poincaréschen “Wiederkehrsatzes”
© Foundation Compositio Mathematica, 1935, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions
A non-conventional ergodic theorem for a nilsystem
  • T. Ziegler
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2005
We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.