Arithmetics in numeration systems with negative quadratic base

Zuzana Masáková; Tomáš Vávra

Displaying similar documents to “Arithmetics in numeration systems with negative quadratic base”

Complexity of infinite words associated with beta-expansions

Christiane Frougny, Zuzana Masáková, Edita Pelantová (2004)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We study the complexity of the infinite word $u_{β}$ associated with the Rényi expansion of $1$ in an irrational base $β > 1$ . When $β$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ (n) = n + 1$ . For $β$ such that $d_{β} (1) = t_{1} t_{2} \dots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_{β}$ . When $t_{m} = 1$ we show that $ℂ (n) = (m - 1) n + 1$ . In the cases when $t_{1} = t_{2} = \dots = t_{m - 1}$ or $t_{1} > max {t_{2}, \dots, t_{m - 1}}$ we show that the first difference of the complexity function $ℂ (n + 1) - ℂ (n)$ takes value in ${m - 1, m}$ for every $n$ , and consequently...

Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers

Lubomíra Balková, Edita Pelantová, Ondřej Turek (2007)

RAIRO - Theoretical Informatics and Applications

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We study some arithmetical and combinatorial properties of -integers for being the larger root of the equation . We determine with the accuracy of 1 the maximal number of -fractional positions, which may arise as a result of addition of two -integers. For the infinite word coding distances between the consecutive -integers, we determine precisely also the balance. The word is the only fixed point of the morphism → and → . In the case , the corresponding infinite word is sturmian,...

A note on univoque self-sturmian numbers

Jean-Paul Allouche (2008)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number $β$ in $(1, 2)$ is univoque and self-sturmian if and only if the $β$ -expansion of $1$ is of the form $1 v$ , where $v$ is a characteristic...

Substitution invariant sturmian bisequences

Bruno Parvaix (1999)

Journal de théorie des nombres de Bordeaux

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We prove that a Sturmian bisequence, with slope $α$ and intercept $ρ$ , is fixed by some non-trivial substitution if and only if $α$ is a Sturm number and $ρ$ belongs to $ℚ (α)$ . We also detail a complementary system of integers connected with Beatty bisequences.

Linear recurrence sequences without zeros

Artūras Dubickas, Aivaras Novikas (2014)

Czechoslovak Mathematical Journal

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Let $a_{d - 1}, \dots, a_{0} \in ℤ$ , where $d \in ℕ$ and $a_{0} \neq 0$ , and let $X = {(x_{n})}_{n = 1}^{\infty}$ be a sequence of integers given by the linear recurrence $x_{n + d} = a_{d - 1} x_{n + d - 1} + \dots + a_{0} x_{n}$ for $n = 1, 2, 3, \dots$ . We show that there are a prime number $p$ and $d$ integers $x_{1}, \dots, x_{d}$ such that no element of the sequence $X = {(x_{n})}_{n = 1}^{\infty}$ defined by the above linear recurrence is divisible by $p$ . Furthermore, for any nonnegative integer $s$ there is a prime number $p \geq 3$ and $d$ integers $x_{1}, \dots, x_{d}$ such that every element of the sequence $X = {(x_{n})}_{n = 1}^{\infty}$ defined as above modulo $p$ belongs to the set ${s + 1, s + 2, \dots, p - s - 1}$ .

Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Luca Q. Zamboni (2000)

Annales de l'institut Fourier

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In a 1982 paper Rauzy showed that the subshift $(X, T)$ generated by the morphism $1 \mapsto 12$ , $2 \mapsto 13$ and $3 \mapsto 1$ is a natural coding of a rotation on the two-dimensional torus $𝕋^{2}$ , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $ℝ^{2},$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2 n + 1$ satisfying a combinatorial criterion known as the $☆$ condition of Arnoux and Rauzy codes the orbit...