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Stanisław Kasjan

Publications and source records attributed to Stanisław Kasjan.

13 recordsLinked to original sources
arXiv2026

On automorphisms of $\mathscr{B}$-admissible and related subshifts

We adapt ideas of Kim and Roush [15], originally developed in the study of automorphisms of sofic subshifts, to obtain sufficient conditions under which a subshift has a huge automorphism group. We apply this approach to non-sofic subshifts defined by sets of multiples. In particular, we establish a dichotomy for the $\mathscr{B}$-admissible subshift: its automorphism group is either trivial or contains an embedded copy of the automorphism group of the full shift $\{0,1\}^{\mathbb Z}$. In the latter case, we say that the automorphism group is huge. We further show that the automorphism group of the hereditary closure of the $\mathscr{B}$-free subshift is huge whenever $\mathscr{B}\subset \mathbb N$ is infinite and contains no infinite pairwise coprime subset.

math.DS↗
arXiv2025

Besicovitch covering numbers for $\mathcal B$-free and other shifts

For a finite alphabet $A$ define by $d_1(x,y):=\limsup_{n\to\infty}\frac{1}{2n+1}\#\{|i|\le n: x_i\neq y_i\}$ the Besicovitch pseudo-metric on $A^{\mathbb Z}$. It is well known that a closed subshift of $A^{\mathbb Z}$ has finite covering numbers w.r.t. $d_1$ if and only if it is mean-equicontinuous. Here we study, more generally, the scaling behavior of these covering numbers for individual orbits which are generic for an ergodic measure $μ$ on $A^{\mathbb Z}$ with discrete spectrum, and we explore their usefulness as invariants for block code equivalence. We illustrate this by developing tools to determine these covering numbers for various classes of $\mathcal B$-free numbers (in particular also for square-free numbers), and we provide a continuous family of measures $μ_s$, all with the same discrete spectrum generated by a single number, but such that $μ_s$- and $μ_{s'}$-typical $x$ resp. $x'\in A^{\mathbb Z}$ have sufficiently different growth of covering numbers such that there are no finite block codes mapping $x\to x'$ and $x'\to x$. (Indeed, both orbits have different amorphic complexities.)

math.DS↗
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.

math.DS↗
arXiv2022

Automorphisms of $\mathcal{B}$-free and other Toeplitz shifts

We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of $\mathcal{B}$-free subshifts satisfying them, extending [10]. On the other hand we provide an example of a $\mathcal{B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 in [13].

math.DS↗
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.

math.DS↗
arXiv2022

Dynamics of $\mathscr{B}$-free systems generated by Behrend sets. I

We study the complexity of $\mathscr{B}$-free subshifts which are proximal and of zero entropy. Such subshifts are generated by Behrend sets. The complexity is shown to achieve any subexponential growth and is estimated for some classical subshifts (prime and semiprime subshifts). We also show that $\mathscr{B}$-admissible subshifts are transitive only for coprime sets $\mathscr{B}$ which allows one to characterize dynamically the subshifts generated by the Erdös sets.

math.DS↗
arXiv2019

On tame strongly simply connected algebras

In this survey we present the criterion for tameness of strongly simply connected algebras due to Brüstle, de la Peña and Skowroński. We recall relevant concepts of representation theory and discuss some applications and connections to other problems.

math.RT↗
arXiv2017

Dynamics of $\mathcal B$-free sets: a view through the window

Let $\mathcal B$ be an infinite subset of $\{1,2,\dots\}$. We characterize arithmetic and dynamical properties of the $\mathcal B$-free set $\mathcal F_{\mathcal B}$ through group theoretical, topological and measure theoretic properties of a set $W$ (called the window) associated with $\mathcal B$. This point of view stems from the interpretation of the set $\mathcal F_{\mathcal B}$ as a weak model set. Our main results are: $\mathcal B$ is taut if and only if the window is Haar regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is proximal if and only if the window has empty interior; and the dynamical system associated to $\mathcal F_{\mathcal B}$ has the "naïvely expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.

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↗
arXiv2014

Polynomial actions of unitary operators and idempotent ultrafilters

Let $p$ be an idempotent ultrafilter over $\mathbb{N}$. For a positive integer $N$, let ${\cal P}_{\leq N}$ denote the additive group of polynomials $P\in\mathbb{Z}[x]$ with ${\rm deg}\, P\leq N$ and $P(0)=0$. Given a unitary operator $U$ on a Hilbert space ${\cal H}$, we prove, for each $N\geq1$, the existence of a unique decomposition ${\cal H}=\bigoplus_{r\geq 1}{\cal H}^{(N)}_r$ into closed, $U$-invariant subspaces such that (a) for any polynomial $P\in{\cal P}_{\leq N}$, we have $$ p\, \text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{P(n)}=0_{{\cal H}_r^{(N)}}\;\mbox{or}\; Id_{{\cal H}_r^{(N)}},\; \mbox{for each}\; r\geq1 ; $$ (b) for each $r\neq s$ there exists $Q\in{\cal P}_{\leq N}$ such that $$ p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{Q(n)}\neq p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_s^{(N)}}\right)^{Q(n)}. $$ In connection with this result we introduce the notion of rigidity group. Namely, a subgroup $G\subset {\cal P}_{\leq N}$ is called an $N$-rigidity group if there exist an idempotent ultrafilter $p$ over $\mathbb{N}$ and a unitary operator $U$ on a Hilbert space $\cal H$ such that $$\label{ab1} G=\{P\in{\cal P}_{\leq N}:\: p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{P(n)}=Id\}$$ and $p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{Q(n)}=0\;\;\mbox{for each}\;\;Q\in{\cal P}_{\leq N}\setminus G.$ The main result of the paper states that a subgroup $G\subset {\cal P}_{\leq N}$ satisfying $\max\{{\rm deg}\, P:\:P\in G\}=N$ is an $N$-rigidity group if and only if $G$ has finite index in ${\cal P}_{\leq N}$.

math.DS↗
arXiv2013

0-1 sequences of the Thue-Morse type and Sarnak's conjecture

We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.

math.DS↗
arXiv2009

On Lie algebras associated with representation directed algebras

Let $B$ be a representation-finite $\mathbb{C}$-algebra. The $\mathbb{Z}$-Lie algebra $L(B)$ associated with $B$ has been defined by Ch. Riedtmann. If $B$ is representation-directed there is another $\mathbb{Z}$-Lie algebra associated with $B$ defined by C. M. Ringel and denoted by $\CK(B)$. We prove that the Lie algebras $L(B)$ and $\CK(B)$ are isomorphic for any representation-directed $\mathbb{C}$-algebra $B$.

math.RT↗