Reversals and palindromes in continued fractions

Adamczewski, Boris; Allouche, Jean‐Paul

doi:10.1016/j.tcs.2007.03.017

Cited by 25 publications

(41 citation statements)

References 59 publications

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“…The number ζ also appears (actually, ζ + 1) as the irrationality measure of numbers of the form (b − 1) n≥1 b − nα [1]. Finally, ζ also appears in a formula giving the critical exponent (aka ''index'') of Sturmian words, as found by Damanik …”

Section: Continued Fraction Expansionsmentioning

confidence: 83%

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Periodicity, repetitions, and orbits of an automatic sequence

Allouche

Rampersad

Shallit

2009

Theoretical Computer Science

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a b s t r a c tWe revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squarefree, or cubefree, etc.). We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope α, have automatic continued fraction expansions if α does.

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Section: Continued Fraction Expansionsmentioning

confidence: 83%

“…The quantity ζ = lim sup n≥0 q n /q n−1 figures in a number of recent papers in combinatorics on words. For example, 2 + ζ is the value of the recurrence quotient of a Sturmian word with slope α [14,1]. Hence this recurrence quotient has a k-automatic continued fraction if α does.…”

Section: Continued Fraction Expansionsmentioning

confidence: 99%

Periodicity, repetitions, and orbits of an automatic sequence

Allouche

Rampersad

Shallit

2009

Theoretical Computer Science

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“…avec a > 1 irrationnel et b f 2 entier, les résultats présentés dans [1] (voir aussi [8]) impliquent que mðxÞ ¼ Dioðx; bÞ ¼ 1 þ lim sup n!þy ½a n ; a nÀ1 ; . .…”

Section: Exposant Diophantien Et Mesures D'irrationalitéunclassified

Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance

Adamczewski

Bugeaud

2011

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…. , a 0 ], the latter computation being done in [1]. Following Section 2.3.6 of [18], a Sturm number is a real number whose continued fraction expansion is either of the form…”

Section: Exponents Of Diophantine Approximation and Maximal Blocksmentioning

confidence: 99%

Morphic and automatic words: maximal blocks and Diophantine approximation

2011

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Let w be a morphic word over a finite alphabet Σ, and let ∆ be a nonempty subset of Σ. We study the behavior of maximal blocks consisting only of letters from ∆ in w, and prove the following: let (i k , j k ) denote the starting and ending positions, respectively, of the k'th maximal ∆-block in w. Then lim sup k→∞ (j k /i k ) is algebraic if w is morphic, and rational if w is automatic. As a result, we show that the same conclusion holds if (i k , j k ) are the starting and ending positions of the k'th maximal zero 1 block, and, more generally, of the k'th maximal x-block, where x is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).

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Reversals and palindromes in continued fractions

Cited by 25 publications

References 59 publications

Periodicity, repetitions, and orbits of an automatic sequence

Periodicity, repetitions, and orbits of an automatic sequence

Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance

Morphic and automatic words: maximal blocks and Diophantine approximation

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