arXiv subjects

Tomáš Vávra

Publications and source records attributed to Tomáš Vávra.

13 recordsLinked to original sources

arXiv2023

Self-similar sets and self-similar measures in the $p$-adics

In this paper we investigate $p$-adic self-similar sets and $p$-adic self-similar measures. We show that $p$-adic self-similar sets are $p$-adic path set fractals, and that the converse is not necessarily true. For $p$-adic self-similar sets and $p$-adic self-similar measures, we show the existence of a unique essential class. We show that, under mild assumptions, the decimation of $p$-adic self-similar sets is maximal. For $p$-adic self-similar measures, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. Most of these results use the additional structure of self-similarity, and are not true in general for $p$-adic path set fractals.

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arXiv2021

Non-expansive matrix number systems with bases similar to $J_n(1)$

We study representations of integral vectors in a number system with a matrix base $M$ and vector digits. We focus on the case when $M$ is similar to $J_n$, the Jordan block of $1$ of size $n$. If $M=J_2$, we classify digit sets of size 2 allowing representation of the whole $\mathbb{Z}^2$. For $J_n$ with $n\geq 3$, it is shown that three digits suffice to represent all of $\mathbb{Z}^n$. For bases similar to $J_n$, at most $n$ digits are required, with the exception of $n=1$. Moreover, the language of strings representing the zero vector with $M=J_2$ and the digits $(0,\pm 1)^T$ is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.

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arXiv2021

On positional representation of integer vectors

We show that any $m\times m$ matrix $M$ with integer entries and $\det M =Δ\neq 0$ can be equipped by a finite digit set $\mathcal{D}\subset\mathbb{Z}^m$ such that any integer $m$-dimensional vector belongs to the set $$ {\rm Fin}_{\mathcal{D}}(M)= \Bigl\{\sum_{k\in I}M^k {d}_k : \emptyset\neq I \text{ finite subset of } \mathbb{Z} \text{ and } {d}_k \in \mathcal{D} \text{ for each } k \in I\Bigr\} \subset \bigcup\limits_{k\in \mathbb{N}} \frac{1}{Δ^k}\mathbb{Z}^{m} \,. $$ We also characterize the matrices $M$ for which the sets $ {\rm Fin}_{\mathcal{D}}(M)$ and $ \bigcup\limits_{k\in \mathbb{N}} \frac{1}{Δ^k}\mathbb{Z}^{m}$ coincide.

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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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arXiv2018

Periodic representations in Salem bases

We prove that all algebraic bases $β$ allow an eventually periodic representations of the elements of $\mathbb Q(β)$ with a finite alphabet of digits $\mathcal A$. Moreover, the classification of bases allowing that those representations have the so-called weak greedy property is given. The decision problem whether a given pair $(β,\mathcal A)$ allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.

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arXiv2017

Periodic representations in algebraic bases

We study periodic representations in number systems with an algebraic base $β$ (not a rational integer). We show that if $β$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such that every element of $\mathbb Q(β)$ admits an eventually periodic representation with base $β$ and digits in $\mathcal A$.

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arXiv2017

Finite beta-expansions with negative bases

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $β$ having the negative finiteness property, that is the set of finite $(-β)$-expansions is equal to $\mathbb{Z}[β^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-β)$-integers. We also give conditions excluding the negative finiteness property.

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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

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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arXiv2015

Pisot unit generators in number fields

Pisot numbers are real algebraic integers bigger than 1, whose other conjugates have all modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular all fields without CM, but not only those.

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arXiv2014

On the Finiteness property of negative cubic Pisot bases

We study arithmetical aspects of Ito-Sadahiro number systems with negative base. We show that the bases $-β<-1$, where $β$ is zero of $x^3-mx^2-mx-m,\ m\in\mathbb N,$ possess the so-called finiteness property. For the Tribonacci base $-γ,$ zero of $x^3-x^2-x-1$, we present an effective algorithm for addition and subtraction. In particular, we present a finite state transducer performing these operations. As a consequence of the structure of the transducer, we determine the maximal number of fractional digits arising from addition or subtraction of two $(-γ)$-integers.

math.CO↗

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.

math.CO↗

arXiv2013

Integers in number systems with positive and negative quadratic Pisot base

We consider numeration systems with base $β$ and $-β$, for quadratic Pisot numbers $β$ and focus on comparing the combinatorial structure of the sets $\Z_β$ and $\Z_{-β}$ of numbers with integer expansion in base $β$, resp. $-β$. Our main result is the comparison of languages of infinite words $u_β$ and $u_{-β}$ coding the ordering of distances between consecutive $β$- and $(-β)$-integers. It turns out that for a class of roots $β$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_β$ can be identified only with the language of a morphic image of $u_{-β}$. We also study the group structure of $(-β)$-integers.

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