arXiv subjects

Bernard Host

Publications and source records attributed to Bernard Host.

30 records · Page 2Linked to original sources

arXiv2009

Nilsequences and a structure theorem for topological dynamical systems

We characterize inverse limits of nilsystems in topological dynamics, via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems. We provide two applications of the structure. The first is to nilsequences, which have played an important role in recent developments in ergodic theory and additive combinatorics; we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally, meaning on finite intervals. The second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system. We show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontin uous factor and corresponds to the case of order 1.

math.DS↗

arXiv2009

Nil-Bohr Sets of Integers

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role in both additive combinatorics and in ergodic theory. Here we introduce a higher order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.

math.DS↗

arXiv2008

Ergodic seminorms for commuting transformations and applications

Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide some tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they will be useful in the solution of other problems.

math.DS↗

arXiv2008

Substitutional dynamical systems, Bratteli diagrams and dimension groups

The present paper explores substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit, and render an effective method to compute an invariant of (ordered) $K$-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background.

math.DS↗

arXiv2008

Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems

The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts and odometers measure--theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. We give partial answers to this question.

math.DS↗

arXiv2007

Uniformity seminorms on $\ell^\infty$ and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Z/N\Z$ introduced by Gowers in his proof of Szemerédi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemerédi's Theorem) defined by the authors. For each integer $k\geq 1$, we define seminorms on $\ell^\infty(\Z)$ analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.

math.DS↗

arXiv2007

Analysis of two step nilsequences

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg's proof of Szemerédi's Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for 2-step nilsequences and give a classification scheme for them.

math.DS↗

arXiv2006

Nilsystèmes d'ordre deux et parallèlépipèdes

A classic family in topological dynamics is that of minimal rotations. One natural extension of this family is the class of nilsystems and their inverse limits. These systems have arisen in recent applications in ergodic theory and in additive combinatorics, renewing interest in studying these classical objects. Minimal rotations can be characterized via the regionally proximal relation. We introduce a new relation, the bi-regionally proximal relation, and show that it characterizes inverse limits of two step nilsystems. Minimal rotations are linked to almost periodic sequences, and more generally nilsystems correspond to nilsequences. Theses sequences were introduced in ergodic theory and have since be used in some questions of Numer Theory. Using our characterization of two step nilsystems we deduce a characterization of two step nilsequences. The proofs rely on in an essential way the study of "parallelepiped structures'' developed by B. Kra and the first author.

math.DS↗

arXiv2006

Progressions arithmétiques dans les nombres premiers, d'après B. Green et T. Tao

B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does not use any "transcendental" method and any of the deep theorems of analytic number theory. It is written in a 'spirit' close to ergodic theory and in particular of Furstenberg's proof of Szemerédi's Theorem, but it does not use any result of this theory. Therefore the method can be considered as elementary, which does not mean easy. We entend here to present the mains ideas of this proof.

math.DS↗

arXiv2006

Multiple recurence and convergence for sequences related to the prime numbers

For any measure preserving system $(X,\mathcal{X},μ,T)$ and $A\in\mathcal{X}$ with $μ(A)>0$, we show that there exist infinitely many primes $p$ such that $μ\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$ (the same holds with $p-1$ replaced by $p+1$). Furthermore, we show the existence of the limit in $L^2(μ)$ of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form $p-1$ (or $p+1$) for some prime $p$.

math.DS↗

arXiv2006

Convergence of multiple ergodic averages

These notes are based on a course for a general audience given at the Centro de Modeliamento Matemático of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times. We also describe the relations between this area of ergodic theory and some classical and some recent results in additive number theory.

math.DS↗

arXiv2006

Parallelepipeds, Nilpotent Groups, and Gowers Norms

In his proof of Szemeredi's Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions 2 and 3 and show when this possible, and describe a correspondence between the parallelepiped structures nilpotent groups.

math.CO↗