arXiv subjects

Zuzana Masáková

Publications and source records attributed to Zuzana Masáková.

At least 19 recordsLinked to original sources

arXiv2026

Parry condition, existence and uniqueness of alternate bases

Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers.

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arXiv2024

Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $β>1$ and derive a necessary condition for validity of the Midy property. For $β=\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property.

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arXiv2024

2-balanced sequences coding rectangle exchange transformation

We define a new class of ternary sequences that are 2-balanced. These sequences are obtained by colouring of Sturmian sequences. We show that the class contains sequences of any given letter frequencies. We provide an upper bound on factor and abelian complexity of these sequences. Using the interpretation by rectangle exchange transformation, we prove that for almost all triples of letter frequencies, the upper bound on factor and abelian complexity is reached. The bound on factor complexity is given using a number-theoretical function which we compute explicitly for a class of parameters.

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arXiv2024

Periodicity and pure periodicity in alternate base systems

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the so-called alternate base $B$ given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. on the set of numbers with eventually periodic $B$-expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic $B$-expansion.

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arXiv2024

Finiteness property in Cantor real numeration systems

For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness property. The proof uses rewriting rules on the~language of~expansions in the corresponding numeration system. The proof is constructive and provides a~method for~performing addition of~expansions in Cantor real bases. We consider a numeration system which is a common generalization of the positional systems introduced by Cantor and Rényi. Number representations are obtained using a composition of $β_k$-transformations for a given sequence of real bases $B=(β_k)_{k\geq 1}$, $β_k>1$. We focus on~arithmetical properties of the set of numbers with finite $B$-expansion in case that $B$ is an alternate base, i.e.\ $B$ is a periodic sequence. We provide necessary conditions for the so-called finiteness property. We further show a~sufficient condition using rewriting rules on the~language of~representations. The proof is constructive and provides a~method for~performing addition of~expansions in alternate bases. Finally, we give a family of alternate bases that satisfy this sufficient condition. Our work generalizes the results of Frougny and Solomyak obtained for the case when the base $B$ is a constant sequence.

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arXiv2023

Substitutions and Cantor real numeration systems

We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence $\Beta=(β_n)_{n\in\Z}$ of real numbers greater than one. We introduce the set of $\Beta$-integers and code the sequence of gaps between consecutive $\Beta$-integers by a symbolic sequence in general over the alphabet $\N$. We show that this sequence is $S$-adic. We focus on alternate base systems, where the sequence $\Beta$ of bases is periodic and characterize alternate bases $\Beta$, in which $\Beta$-integers can be coded using a symbolic sequence $v_{\Beta}$ over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that $v_\Beta$ is a fixed point of their composition. The paper generalizes results of Fabre and Burdík et al.\ obtained for the Rényi numerations systems, i.e., in the case when the Cantor base $\Beta$ is a constant sequence.

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arXiv2022

Spectrum, algebraicity and normalization in alternate bases

The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbolβ=(β_0,\dots,β_{p-1})$ determining sofic systems. We show that a necessary condition is that the product $δ=\prod_{i=0}^{p-1}β_i$ is an algebraic integer and all of the bases $β_0,\ldots,β_{p-1}$ belong to the algebraic field ${\mathbb Q}(δ)$. On the other hand, we also give a sufficient condition: if $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, then the system associated with the alternate base $\boldsymbolβ=(β_0,\dots,β_{p-1})$ is sofic. The second aim of this paper is to provide an analogy of Frougny's result concerning normalization of real bases representations. We show that given an alternate base $\boldsymbolβ=(β_0,\dots,β_{p-1})$ such that $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. The spectrum of a real number $δ>1$ and an alphabet $A\subset {\mathbb Z}$ was introduced by Erdős et al. For our purposes, we use a generalized concept with $δ\in{\mathbb C}$ and $A\subset{\mathbb C}$ and study its topological properties.

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arXiv2020

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

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arXiv2020

Finiteness and periodicity of continued fractions over quadratic number fields

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Perron numbers $β$, the $β$-continued fraction represents finitely all elements of the quadratic field $\mathbb Q(β)$, thus answering questions of Rosen and Bernat. Based on the validity of a conjecture of Mercat, these are all quadratic Perron numbers with this feature.

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arXiv2020

On generalized self-similarities of cut-and-project sets

Cut-and-project sets $Σ\subset\mathbb{R}^n$ represent one of the types of uniformly discrete relatively dense sets. They arise by projection of a section of a higher-dimensional lattice to a suitably oriented subspace. Cut-and-project sets find application in solid state physics as mathematical models of atomic positions in quasicrystals, the description of their symmetries is therefore of high importance. We focus on the question when a linear map $A$ on $\mathbb{R}^n$ is a self-similarity of a cut-and-project set $Σ$, i.e.\ satisfies $AΣ\subsetΣ$. We characterize such mappings $A$ and provide a construction of a suitable cut-and-project set $Σ$. We determine minimal dimension of a lattice which permits construction of such a set $Σ$.

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arXiv2019

Morphisms generating antipalindromic words

We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism $\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two classes contain all morphisms (up to conjugation) which generate infinite words with infinitely many antipalindromes. This is an analogue to the famous HKS conjecture concerning infinite words containing infinitely many palindromes. We prove our conjecture for two special classes of morphisms, namely (i) uniform morphisms and (ii) morphisms with fixed points containing also infinitely many palindromes.

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arXiv2018

Palindromic length of words and morphisms in class $\mathcal{P}$

We study the palindromic length of factors of infinite words fixed by morphisms of the so-called class $\mathcal{P}$ introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue-Morse word we provide estimates on the constants of the growth. We also construct an infinite word rich in palindromes for which the palindromic length grows as $\sqrt{n}$.

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arXiv2016

On the spectra of Pisot-cyclotomic numbers

We investigate the complex spectra \[ X^{\mathcal A}(β)=\left\{\sum_{j=0}^na_jβ^j : n\in{\mathbb N},\ a_j\in{\mathcal A}\right\} \] where $β$ is a quadratic or cubic Pisot-cyclotomic number and the alphabet $\mathcal A$ is given by $0$ along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut and project method.

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arXiv2016

Exchange of three intervals: itineraries, substitutions and palindromicity

Given a symmetric exchange of three intervals, we provide a detailed description of the return times to a subinterval and the corresponding itineraries. We apply our results to morphisms fixing words coding non-degenerate three interval exchange transformation. This allows us to prove that the conjecture stated by Hof, Knill and Simon is valid for such infinite words.

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arXiv2016

On periodic representations in non-Pisot bases

We study periodic expansions in positional number systems with a base $β\in\C,\ |β|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(β),$ such that all elements of $\Q(β)$ admit at least one eventually periodic representation with digits in $\A$. In this paper we prove a general result that guarantees the existence of such an $\A$. This result implies the existence of such an $\A$ when $β$ is a rational number or an algebraic integer with no conjugates of modulus $1$. We also consider eventually periodic representations of elements of $\Q(β)$ for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of $\Q(β)$ admits such a representation then $β$ must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt \cite{Schmidt}.

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arXiv2014

On distinct unit generated fields that are totally complex

We consider the problem of characterizing all number fields $K$ such that all algebraic integers $α\in K$ can be written as the sum of distinct units of $K$. We extend a method due to Thuswaldner and Ziegler that previously did not work for totally complex fields and apply our results to the case of totally complex quartic number fields.

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arXiv2014

Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

In this paper we study the expansions of real numbers in positive and negative real base as introduced by Rényi, and Ito & Sadahiro, respectively. In particular, we compare the sets $\mathbb{Z}_β^+$ and $\mathbb{Z}_{-β}$ of nonnegative $β$-integers and $(-β)$-integers. We describe all bases $(\pmβ)$ for which $\mathbb{Z}_β^+$ and $\mathbb{Z}_{-β}$ can be coded by infinite words which are fixed points of conjugated morphisms, and consequently have the same language. Moreover, we prove that this happens precisely for $β$ with another interesting property, namely that any integer linear combination of non-negative powers of the base $-β$ with coefficients in $\{0,1,\dots,\lfloorβ\rfloor\}$ is a $(-β)$-integer, although the corresponding sequence of digits is forbidden as a $(-β)$-integer.

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arXiv2014

Description of spectra of quadratic Pisot units

The spectrum of a real number $β>1$ is the set $X^{m}(β)$ of $p(β)$ where $p$ ranges over all polynomials with coefficients restricted to ${\mathcal A}=\{0,1,\dots,m\}$. For a quadratic Pisot unit $β$, we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spectra in the frame of the cut-and-project scheme. We also show that shifting the set ${\mathcal A}$ of digits so that it contains at least one negative element, or considering negative base $-β$ instead of $β$, the gap sequence of the generalized spectrum is a coding of an exchange of three intervals.

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