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Joanna Kułaga-Przymus

Publications and source records attributed to Joanna Kułaga-Przymus.

At least 19 recordsLinked to original sources
arXiv2026

$\mathfrak{B}$-free integers in number fields and dynamics

In 2010, Sarnak initiated the study of the dynamics of the system determined by the square of the Möbius function (the characteristic function of the square-free integers). We deal with his program in the more general context of $\mathfrak{B}$-free integers in number fields, suggested 5 years later by Baake and Huck. This setting encompasses the classical square-free case and its generalizations. Given a number field $K$, let $\mathfrak{B}$ be a family of pairwise coprime ideals in its ring of integers $\mathcal{O}_K$, such that $\sum_{\mathfrak{b}\in\mathfrak{B}}1/|\mathcal{O}_K / \mathfrak{b}|<\infty$. We study the dynamical system determined by the set $\mathcal{F}_\mathfrak{B}=\mathcal{O}_K\setminus \bigcup_{\mathfrak{b}\in\mathfrak{B}}\mathfrak{b}$ of $\mathfrak{B}$-free integers in $\mathcal{O}_K$. We show that the characteristic function $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$ of $\mathcal{F}_\mathfrak{B}$ is generic along the natural Følner sequence for a probability measure on $\{0,1\}^{\mathcal{O}_K}$, invariant under the multidimensional shift. The corresponding measure-theoretical dynamical system is proved to be isomorphic to an ergodic rotation on a compact Abelian group. In particular, it is of zero Kolmogorov entropy. Moreover, we provide a description of ``patterns'' appearing in $\mathcal{F}_\mathfrak{B}$ and compute the topological entropy of the orbit closure of $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$. Finally, we show that this topological dynamical system has a non-trivial topological joining with an ergodic rotation on a compact Abelian group.

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arXiv2025

A note on $\mathscr{B}$-free sets and the existence of natural density

Given $\mathscr{B}\subseteq \mathbb{N}$, let $\mathcal{M}_\mathscr{B}=\bigcup_{b\in\mathscr{B}}b\mathbb{Z}$ be the correspoding set of multiples. We say that $\mathscr{B}$ is taut if the logarithmic density of $\mathcal{M}_\mathscr{B}$ decreases after removing any element from $\mathscr{B}$. We say that $\mathscr{B}$ is minimal if it is primitive (i.e.\ $b| b'$ for $b,b'\in\mathscr{B}$ implies $b=b'$) and the characteristic function $η$ of $\mathcal{M}_\mathscr{B}$ is a Toeplitz sequence (i.e.\ for every $n\in \mathbb{N}$ there exists $s_n$ such that $η$ is constant along $n+s_n\mathbb{Z}$). With every $\mathscr{B}$ one associates the corresponding taut set $\mathscr{B}'$ (determined uniquely among all taut sets by the condition that the associated Mirsky measures agree) and the minimal set $\mathscr{B}^*$ (determined uniquely among all minimal sets by the condition that every configuration appearing on $\mathcal{M}_{\mathscr{B}^*}$ appears on $\mathcal{M}_\mathscr{B}$: for every $n\in \mathbb{N}$, there exists $k\in \mathbb{Z}$ such that $\mathcal{M}_{\mathscr{B}^*}\cap [0,n]=\mathcal{M}_\mathscr{B} \cap[k,k+n]-k$). Besicovitch [2] gave an example of $\mathscr{B}$ whose set of multiples does not have the natural density. It was proved in [7, Lemma 4.18] that if $\mathcal{M}_{\mathscr{B}'}$ posses the natural density then so does $\mathcal{M}_\mathscr{B}$. In this paper we show that this is the only obstruction: every configuration $ijk\in \{0,1\}^3$ (with $ij\neq 01$), encoding the information on the existence of the natural density for the triple $\mathcal{M}_\mathscr{B},\mathcal{M}_{\mathscr{B'}},\mathcal{M}_{\mathscr{B}^*}$, can occur. Furthermore, we show that $\mathcal{M}_\mathscr{B}$ and $\mathcal{M}_{\mathscr{B}'}$ can differ along a set of positive upper density.

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arXiv2024

Hereditary subshifts whose measure of maximal entropy has no Gibbs property

We show that the measure of maximal entropy for the hereditary closure of a $\mathscr{B}$-free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, we show that $\mathscr{B}$ is taut whenever the corresponding Mirsky measure $ν_η$ has full support. This is the converse theorem to a recent result of Keller.

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arXiv2024

Invariant measures for $\mathscr{B}$-free systems revisited

For $ \mathscr{B} \subseteq \mathbb{N} $, the $ \mathscr{B} $-free subshift $ X_η $ is the orbit closure of the characteristic function of the set of $ \mathscr{B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_η $, have their analogues for $ X_η $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $ η^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_η $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_η $ from above and below.

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arXiv2024

Basic thermodynamical formalism for sandwich subshifts

Consider a partial order on $\{0,1\}^{\mathbb Z}: x\leq y$ when $x_i\leq y_i$ for all $i\in\mathbb{Z}$. A subshift $X\subset\{0,1\}^{\mathbb{Z}}$ is hereditary if together with any $x\in \{0,1\}^{\mathbb Z}$ it contains all $y\leq x$. Heuristically speaking, a hereditary subshift contains all the elements between maximal elements (with respect to this partial order) and the element $0^{\mathbb Z}$. In a particular situation when it suffices to take (the orbit closure of) all the elements between a single maximal element $x$ and the element $0^{\mathbb Z}$, we speak of subordinate subshifts. In this paper we investigate measure-theoretic properties of such subshifts, with a special emphasis on thermodynamical formalism. The key notion is a measure-theoretic counterpart of subordinate subshifts, where the role of a single maximal element is replaced with a single (maximal with respect to a certain order) invariant measure on $\{0,1\}^{\mathbb{Z}}$. We also introduce and investigate two-sided analogues of the above classes, we call them {\it sandwich hereditary}, {\it sandwich subordinate} and {\it sandwich measure-theoretically subordinate} subshifts. Sandwich hereditary subshifts can be thought of as sets of elements between some pairs of maximal and minimal elements satisfying certain assumptions. Sandwich subordinate subshifts occur when it suffices to take (the orbit closure of) all the elements between a single pair of sequences $(w,x)$, where $w\leq x$. In sandwich measure-theoretically subordinate subshifts, the role of a pair of sequences is replaced by a pair (precisely speaking: a joining) of two invariant measures on $\{0,1\}^{\mathbb{Z}}$. The notions and results are motivated by those from the theory of so-called $\mathscr{B}$-free systems.

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arXiv2022

Minimality of $\mathfrak{B}$-free systems in number fields

Let $K$ be a finite extension of $\mathbb{Q}$ and $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{B}$ be a primitive collection of ideals in $\mathcal{O}_K$. We show that any $\mathfrak{B}$-free system is essentially minimal. Moreoever, the $\mathfrak{B}$-free system is minimal if and only if the characteristic function of $\mathfrak{B}$-free numbers is a Toeplitz sequence. Equivalently, there are no ideal $\mathfrak{d}$ and no infinite pairwise coprime collection of ideals $\mathcal{C}$ such that $\mathfrak{d}\mathcal{C}\subseteq\mathfrak{B}$. Moreover, we find a periodic structure in the Toeplitz case. Last but not least, we describe the restrictions on the cosets of ideals contained in unions of ideals.

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arXiv2021

Entropy rate of product of independent processes

We study the multiplicative version of the classical Furstenberg's filtering problem, where instead of the sum $\mathbf{X}+\mathbf{Y}$ one considers the product $\mathbf{X}\cdot \mathbf{Y}$ ($\mathbf{X}$ and $\mathbf{Y}$ are bilateral, real, finitely-valued, stationary independent processes, $\mathbf{Y}$ is taking values in $\{0,1\}$). We provide formulas for $\mathbf{H}(\mathbf{X}\cdot\mathbf{Y}|\mathbf{Y})$. As a consequence, we show that if $\mathbf{H}(\mathbf{X})>\mathbf{H}(\mathbf{Y})=0$ and $\mathbf{X}\amalg \mathbf{Y}$, then $\mathbf{H}(\mathbf{X}\cdot \mathbf{Y})<\mathbf{H}(\mathbf{X})$ (and thus $\mathbf{X}$ cannot be filtered out from $\mathbf{X}\cdot\mathbf{Y}$) whenever $\mathbf{X}$ is not bilaterally deterministic, $\mathbf{Y}$ is ergodic and $\mathbf{Y}$ first return to $1$ can take arbitrarily long with positive probability. On the other hand, if almost surely $\mathbf{Y}$ visits $1$ along an infinite arithmetic progression of a fixed difference (with possibly some more visits in between) then we can find $\mathbf{X}$ that is not bilaterally deterministic and such that $\mathbf{H}(\mathbf{X}\cdot\mathbf{Y})=\mathbf{H}(\mathbf{X})$. As a consequence, a $\mathscr{B}$-free system $(X_η,S)$ is proximal if and only if there is always an entropy drop $h(κ\astν_η)<h(κ)$ for any $κ$ corresponding to a non-bilaterally deterministic process of positive entropy. These results partly settle some open problems on invariant measures for $\mathscr{B}$-free systems.

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arXiv2019

A structure theorem for level sets of multiplicative functions and applications

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost periodic and a pseudo-random parts. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. Let $E=\{n_1 0$, all $\ell\geq 1$ and all polynomials $p_i\in\mathbb{Z}[x]$, $i=1,\ldots,\ell$, with $p_i(0)=0$ we have $$ \lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N μ\big(A\cap T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$ We also show that if a level set $E$ of a multiplicative function has positive upper density, then any self-shift $E-r$, $r\in E$, is a set of averaging polynomial multiple recurrence. This in turn leads to the following refinement of the polynomial Szemerédi theorem (cf. [4]). Let $E$ be a level set of an arbitrary multiplicative function, suppose $E$ has positive upper density and let $r\in E$. Then for any set $D\subset \mathbb{N}$ with positive upper density and any polynomials $p_i\in\mathbb{Q}[t]$, $i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $β>0$ such that the set $$ \left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>β\,\right\} $$ has positive lower density.

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arXiv2018

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},κ,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbolμ$ in all zero entropy systems, in particular, it makes $\boldsymbolμ$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.

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arXiv2018

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set $R\subset \mathbb{N}$ is called rational if it is well-approximable by finite unions of arithmetic progressions. Examples 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 $Φ_x:=\{n\in\mathbb{N}: \frac{\boldsymbolφ(n)}{n} 0$, then the following are equivalent: (a) $R$ is divisible, i.e. $\overline{d}(R\cap u \mathbb{N})>0$ for all $u\in\mathbb{N}$. (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 $R$ is rational and divisible, then for any set $E\subset\mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_i\in\mathbb{Q}[t]$,$i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $β>0$ such that the set $$\{n\in R:\overline{d}( E\cap (E-p_1(n))\cap\ldots\cap(E-p_\ell(n)))>β\}$$ has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences. We prove that if $\mathcal{A}$ is a finite alphabet, $η\in\mathcal{A}^\mathbb{N}$ is rationally almost periodic, $S$ denotes the left-shift on $\mathcal{A}^\mathbb{Z}$ and $$X:=\{y\in \mathcal{A}^\mathbb{Z} : \text{each finite word appearing in $y$ appears in }η\},$$ then $η$ is a generic point for an $S$-invariant probability measure $ν$ on $X$ such that $(X,ν,S)$ is ergodic and has rational discrete spectrum.

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arXiv2017

Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's multiple mixing question in this context. The main tool is a variation of the Ratner property (a property originally proved by Ratner for the horocycle flow), i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behaviour of these flows and has implications to joinings classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski's main results, by showing that Arnold's flows are mixing of all orders for almost every location of the singularities.

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arXiv2017

Möbius disjointness along ergodic sequences for uniquely ergodic actions

We show that there are an irrational rotation $Tx=x+α$ on the circle $\mathbb{T}$ and a continuous $φ\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{φ,\mathcal{S}}$ acting on $(X\times Y,μ\otimesν)$ by the formula $T_{φ,\mathcal{S}}(x,y)=(Tx,S_{φ(x)}(y))$, where $μ$ stands for Lebesgue measure on $\mathbb{T}$ and $ν$ denotes the unique $\mathcal{S}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak's conjecture on Möbius disjointness holds for all uniquely ergodic models of $T_{φ,\mathcal{S}}$. Moreover, we obtain a class of "random" ergodic sequences $(c_n)\subset\mathbb{Z}$ such that if $\boldsymbolμ$ denotes the Möbius function, then $$ \lim_{N\to\infty}\frac1N\sum_{n\leq N}g(S_{c_n}y)\boldsymbolμ(n)=0 $$ for all (continuous) uniquely ergodic flows $\mathcal{S}$, all $g\in C(Y)$ and $y\in Y$.

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arXiv2016

Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Let $ G $ be a connected, simply connected nilpotent Lie group and $ Γ< G $ a lattice. We prove that each ergodic diffeomorphism $ ϕ(xΓ)=uA(x)Γ$ on the nilmanifold $ G/Γ$, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(Γ)=Γ$, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on Möbius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Möbius orthogonality holds on so called typical short interval: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(ϕ^n(xΓ))μ(n)\right|\to 0$ as $ H\to\infty $ and $ H/M\to0 $ for each $ xΓ\in G/Γ$ and each $ f\in C(G/Γ) $. In particular, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function $μ$ on a typical short interval. We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.

math.DS↗
arXiv2015

$\mathscr{B}$-free sets and dynamics

Let $B\subset \mathbb{N}$ and let $η\in \{0,1\}^\mathbb{Z}$ be the characteristic function of the set $F_B:=\mathbb{Z}\setminus\bigcup_{b}b\mathbb{Z}$ of B-free numbers. Consider $(S,X_η)$, where $X_η$ is the closure of the orbit of $η$ under the left shift S. When $B=\{p^2 : p\in P\}$, $(S,X_η)$ was studied by Sarnak. This case + some generalizations, including the case (*) of B infinite, coprime with $\sum_{b}1/b<\infty$, were discussed by several authors. For general B, contrary to (*), we may have $X_η\subsetneq X_B:=\{x\in \{0,1\}^\mathbb{Z} : |\text{supp }x\bmod b|\leq b-1 \forall_b\}$. Also, $X_η$ may not be hereditary (heredity means that if $x\in X$ and $y\leq x$ coordinatewise then $y\in X$). We show that $η$ is quasi-generic for a natural measure $ν_η$. We solve the problem of proximality by showing first that $X_η$ has a unique minimal (Toeplitz) subsystem. Moreover B-free system is proximal iff B contains an infinite coprime set. B is taut when $δ(F_B)<δ(F_{B\setminus \{b\} })$ for each b. We give a characterization of taut B in terms of the support of $ν_η$. Moreover, for any B there exists a taut B' with $ν_η=ν_{η'}$. For taut sets B,B', we have B=B' iff $X_B=X_{B'}$. For each B there is a taut B' with $\tilde{X}_{η'}\subset \tilde{X}_η$ and all invariant measures for $(S,\tilde{X}_η)$ live on $\tilde{X}_{η'}$. $(S,\tilde{X}_η)$ is shown to be intrinsically ergodic for all B. We give a description of all invariant measures for $(S,\tilde{X}_η)$. The topological entropies of $(S,\tilde{X}_η)$ and $(S,X_B)$ are both equal to $\overline{d}(F_B)$. We show that for a subclass of taut B-free systems proximality is the same as heredity. Finally, we give applications in number theory on gaps between consecutive B-free numbers. We apply our results to the set of abundant numbers.

math.DS↗
arXiv2015

Hereditary subshifts whose simplex of invariant measures is Poulsen

We give a sufficient condition for the simplex of invariant measures for a hereditary system to be Poulsen. In particular, we show that this simplex is Poulsen in case of positive entropy $\mathscr{B}$-free systems. We also give an example of a positive entropy hereditary system whose simplex of invariant measures is not Poulsen.

math.DS↗
arXiv2014

On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

We compare self-joining- and embeddability properties. In particular, we prove that a measure preserving flow $(T_t)_{t\in\mathbb{R}}$ with $T_1$ ergodic is 2-fold quasi-simple (2-fold distally simple) if and only if $T_1$ is 2-fold quasi-simple (2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow $(T_t)_{t\in\mathbb{R}}$ with $T_1$ ergodic with respect to any flow factor is the same for $(T_t)_{t\in\mathbb{R}}$ and for $T_1$. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one map is simple. We describe two classes of flows (flows with minimal self-joining property and flows with the so-called Ratner property) whose time-one maps have unique embeddings into measurable flows. We also give an example of a 2-fold simple flow whose time-one map has more than one embedding.

math.DS↗
arXiv2014

The Moebius function and continuous extensions of rotations

Let $f\colon \mathbb{T}\to \mathbb{R}$ be of class $C^{1+δ}$ for some $δ>0$ and let $c\in\mathbb{Z}$. We show that for a generic $α\in\mathbb{R}$, the extension $T_{c,f}\colon \mathbb{T}^2\to\mathbb{T}^2$ of the irrational rotation $Tx=x+α$, given by $T_{c,f}(x,u)=(x+α, u+cx+f(x))$ ($\bmod\ 1$) satisfies Sarnak's conjecture.

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