A Dynamical Point of View on the Set of<i>ℬ</i>-Free Integers

Abdalaoui, El Houcein El; Lemańczyk, Mariusz; Rue, Thierry de la

doi:10.1093/imrn/rnu164

Cited by 46 publications

(99 citation statements)

References 9 publications

(12 reference statements)

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“…η(n) = 1 if and only if n ∈ F B , and consider the orbit closure X η of η in the shift dynamical system ({0, 1} , σ), where σ stands for the left shift. Topological dynamics and ergodic theory provide a wealth of concepts to describe various aspects of the structure of η, see [11] which originated this point of view by studying the set of square-free numbers, and also [1], [2], [7], [8] which continued this line of research. We collect some facts from these references: (E) If B has light tails and if B contains an infinite pairwise coprime subset, then X η is hereditary,…”

Section: Consequences For the Dynamics Of B-free Systemsmentioning

confidence: 99%

Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics

Keller

2019

Studia Math.

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For any set B ⊆ AE = {1, 2, . . . } one can define its set of multiples M B := b∈B b and the set of B-free numbers F B := \ M B . Tautness of the set B is a basic property related to questions around the asymptotic density of M B ⊆ . From a dynamical systems point of view (originated in [11]) one studies η, the indicator function of F B ⊆ , its shift-orbit closure X η ⊆ {0, 1} and the stationary probability measure ν η defined on X η by the frequencies of finite blocks in η. In this paper we prove that tautness implies the following two properties of η: -The measure ν η has full topological support in X η .-If X η is proximal, i.e. if the one-point set {. . . 000 . . . } is contained in X η and is the unique minimal subset of X η , then X η is hereditary, i.e. if x ∈ X η and if w is an arbitrary element of {0, 1} , then also the coordinate-wise product w · x belongs to X η . This strengthens two results from [2] which need the stronger assumption that B has light tails for the same conclusions.MSC 2010 clasification: 37A35, 37A45, 37B05.

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Section: Consequences For the Dynamics Of B-free Systemsmentioning

confidence: 99%

Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics

Keller

2019

Studia Math.

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“…• Rational subshifts can have positive topological entropy (for the definition of topological entropy see for instance [54, Section 6.3]); for example, rational B-free subshifts can have positive topological entropy, see [4,28,53]. Moreover, they can have many invariant measures [43].…”

Section: Definition Of Rational Subshifts Examplesmentioning

confidence: 99%

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Bergelson¹,

Kułaga-Przymus²,

Lemańczyk³

et al. 2018

Ergod. Th. Dynam. Sys.

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A set R ⊂ N is called rational if it is well-approximable by finite unions of arithmetic progressions, meaning that for every ǫ > 0 there exists a set B = r i=1 a i N+b i , where a 1 , . . . , a r , b 1 , . . . , b r ∈ N, such thatExamples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form Φ x := {n ∈ N :< x}, where x ∈ [0, 1] and ϕ is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if R ⊂ N is a rational set with d(R) > 0, then the following are equivalent:(b) R is an averaging set of polynomial single recurrence.(c) R is an averaging set of polynomial multiple recurrence.As an application, we show that if R ⊂ N is rational and divisible, then for any set E ⊂ N with d(E) > 0 and any polynomials p i ∈ Q[t], i = 1, . . . , ℓ, which satisfy p i (Z) ⊂ Z and p i (0) = 0 for all i ∈ {1, . . . , ℓ}, there exists β > 0 such that the sethas positive lower density.Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if A is a finite alphabet, η ∈ A N is rationally almost periodic, S denotes the left-shift on A Z and X := {y ∈ A Z : each finite word appearing in y appears in η}, then η is a generic point for an S-invariant probability measure ν on X such that the measure preserving system (X, ν, S) is ergodic and has rational discrete spectrum.

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“…10.4]. As such, it is a special case within the larger class of B-free shifts; see [49] and references therein, and [29] for general background. Now, it is shown in [49] that X sf has minimal symmetry group, and it is also clear that X sf is reflection invariant (because R(x sf ) = x sf ), so we are once again in the standard situation of Fact 1 with R(X sf ) = S ⋊ R ≃ Z ⋊ C 2 .…”

Section: Binary Shiftsmentioning

confidence: 99%

Reversing and extended symmetries of shift spaces

Baake¹,

Roberts²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue-Morse system with various generalisations or the Rudin-Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of extended symmetries. We develop their basic theory for faithful Z d -actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier's shift, which is an example of algebraic origin.

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A Dynamical Point of View on the Set ofℬ-Free Integers

Cited by 46 publications

References 9 publications

Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics

Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Reversing and extended symmetries of shift spaces

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