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 ≥ 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 ≥ 2 , we state that a multidimensional infinite word x : ℕ d → 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 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...

(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 ∑ n = 1 ∞ λ ( n ) X n ∈ 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.