Cobham’s theorem and its extensions

Jason P. Bell

Displaying similar documents to “Cobham’s theorem and its extensions”

A Characterization of Multidimensional $S$ -Automatic Sequences

Emilie Charlier, Tomi Kärki, Michel Rigo (2009)

Actes des rencontres du CIRM

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An infinite word is $S$ -automatic if, for all $n \geq 0$ , its $(n + 1)$ st letter is the output of a deterministic automaton fed with the representation of $n$ in the considered numeration system $S$ . In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for $d \geq 2$ , we state that a multidimensional infinite word $x : ℕ^{d} \to Σ$ over a finite alphabet $Σ$ is $S$ -automatic for some abstract...

On gaps in Rényi $β$ -expansions of unity for $β > 1$ an algebraic number

Jean-Louis Verger-Gaugry (2006)

Annales de l’institut Fourier

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Let $β > 1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $β$ -expansion $d_{β} (1)$ of unity which controls the set $ℤ_{β}$ of $β$ -integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{β} (1)$ are shown to exhibit a “gappiness” asymptotically bounded above by $log (M (β)) / log (β)$ , where $M (β)$ is the Mahler measure of $β$ . The proof of this result provides in a natural way a new classification of algebraic numbers $> 1$ with classes...

Substitutions with Cofinal Fixed Points

Bo TAN, Zhi-Xiong WEN, Jun WU, Zhi-Ying WEN (2006)

Annales de l’institut Fourier

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Let $ϕ$ be a substitution over a 2-letter alphabet, say ${a, b}$ . If $ϕ (a)$ and $ϕ (b)$ begin with $a$ and $b$ respectively, $ϕ$ has two fixed points beginning with $a$ and $b$ respectively. We characterize substitutions with two cofinal fixed points (i.e., which differ only by prefixes). The proof is a combinatorial one, based on the study of repetitions of words in the fixed points.

Subword complexity and finite characteristic numbers

Alina Firicel (2009)

Actes des rencontres du CIRM

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Decimal expansions of classical constants such as $\sqrt{2}$ , $π$ and $ζ (3)$ have long been a source of difficult questions. In the case of finite characteristic numbers (Laurent series with coefficients in a finite field), where no carry-over difficulties appear, the situation seems to be simplified and drastically different. On the other hand, the theory of Drinfeld modules provides analogs of real numbers such as $π$ , $e$ or $ζ$ values. Hence, it became reasonable to enquire how “complex” the Laurent representation...

Finite automata and arithmetic.

Allouche, Jean-Paul (1993)

Séminaire Lotharingien de Combinatoire [electronic only]

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(Non)Automaticity of number theoretic functions

Michael Coons (2010)

Journal de Théorie des Nombres de Bordeaux

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Denote by $λ (n)$ Liouville’s function concerning the parity of the number of prime divisors of $n$ . Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that $λ (n)$ is not $k$ –automatic for any $k > 2$ . This yields that $\sum_{n = 1}^{\infty} λ (n) X^{n} \in 𝔽_{p} [[X]]$ is transcendental over $𝔽_{p} (X)$ for any prime $p > 2$ . Similar results are proven (or reproven) for many common number–theoretic functions, including $ϕ$ , $μ$ , $Ω$ , $ω$ , $ρ$ , and others.

Displaying similar documents to “Cobham’s theorem and its extensions”

A Characterization of Multidimensional S -Automatic Sequences

On gaps in Rényi β -expansions of unity for β &gt; 1 an algebraic number

Substitutions with Cofinal Fixed Points

Subword complexity and finite characteristic numbers

Finite automata and arithmetic.

(Non)Automaticity of number theoretic functions

A Characterization of Multidimensional $S$ -Automatic Sequences

On gaps in Rényi $β$ -expansions of unity for $β > 1$ an algebraic number