IPr⁎-recurrence and nilsystems

  title={IPr⁎-recurrence and nilsystems},
  author={Vitaly Bergelson and Alexander Leibman},
  journal={Advances in Mathematics},
8 Citations
Polynomial orbits in totally minimal systems
Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye [13], we prove that the maximal ∞-step pro-nilfactor X∞ of a minimal system (X ,T ) is the topological characteristic factor along
Multiplicative combinatorial properties of return time sets in minimal dynamical systems
We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those
Ju l 2 02 1 Polynomial multiple recurrence and large intersections in rings of integers
We establish characteristic factors for natural classes of polynomial multiple ergodic averages in rings of integers and derive corresponding Khintchine-type recurrence theorems, extending results of
Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets
In the topological dynamical system (X,T ), a point x simultaneously approximates a point y if there exists a sequence n1, n2, . . . of natural numbers for which T ix, T ix, . . . , T ix all tend to
Popular differences for polynomial patterns in rings of integers
We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is
Bracket words: a generalisation of Sturmian words arising from generalised polynomials
Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many
Sublacunary sets and interpolation sets for nilsequences
  • A. Le
  • Computer Science
    Discrete & Continuous Dynamical Systems
  • 2021
A new class of interpolation sets for Bohr almost periodic sequences is provided, and it is proved that the union of an interpolation set for nilsequences and a finite set is an interpolations set fornilsequences.
Sets of large values of correlation functions for polynomial cubic configurations
We prove that for any set $E\subseteq \mathbb{Z}$ with upper Banach density $d^{\ast }(E)>0$ , the set ‘of cubic configurations’ in $E$ is large in the following sense: for any $k\in \mathbb{N}$ and


Ultrafilters, IP sets, Dynamics, and Combinatorial Number Theory
We survey the connection between ultrafilters, ergodic theory, and combinatorics.
Infinite-step nilsystems, independence and complexity
Abstract An ∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an ∞-step nilsystem if and only if it has no non-trivial pairs
Anzai and Furstenberg Transformations on the 2-Torus and Topologically Quasi-Discrete Spectrum
  • K. Kodaka
  • Mathematics
    Canadian Mathematical Bulletin
  • 1995
Abstract Let ϕ0 be an Anzai transformation on the 2-torus T2 defined by ϕ0(x,y) = (e2πiθx,xy) and ϕy a Furstenberg transformation on T2 defined by ϕf(x,y) = (e2πiθx,e2πif(x)xy) where θ is an
Reflections on equicontinuity
We study different conditions which turn out to be equivalent to equicontinuity for a transitive compact Hausdorff flow with a general group action. Among them are a notion of "regional"
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
Some counterexamples in topological dynamics
  • R. Pavlov
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2008
Abstract In this paper, we exhibit, for any sparse-enough increasing sequence {pn} of integers, totally minimal, totally uniquely ergodic, and topologically mixing systems (X,T) and (X′,T′) and
The Mobius function is strongly orthogonal to nilsequences
We show that the Mobius function (n) is strongly asymptotically or- thogonal to any polynomial nilsequence (F (g(n))) n2N. Here, G is a sim- ply-connected nilpotent Lie group with a discrete and
A study of the proximal relation in coset transformation groups
1. Introduction. In this paper, we investigate the proximal relation and related notions in coset transformation groups. In ?1, we show that many of the interesting properties of the proximal
Nil Bohr-sets and almost automorphy of higher order
Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can