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Displaying similar documents to “Subword complexity and finite characteristic numbers”

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...

Cobham’s theorem and its extensions

Jason P. Bell (2009)

Actes des rencontres du CIRM

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Cobham’s theorem says that if k and ℓ are two multiplicatively independent integers and f ( n ) is a k - and ℓ -automatic sequence, then f ( n ) is eventually periodic. We give a summary of recent work on automatic sequences and their relation to Cobham’s theorem.

(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.

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...