Extremal properties of (epi)Sturmian sequences and distribution modulo $1$

Allouche, Jean‐Paul; Glen, Amy

doi:10.4171/lem/56-3-5

Cited by 10 publications

(39 citation statements)

References 71 publications

(173 reference statements)

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“…and hence Ψ z (a)s l Ψ z (x) ∈ F(t) since z is separating for t. So, if x = z then as (1) m−1 x ∈ F(t (1) ), which is impossible; whence x = z. But then, s (1) m−1 x) is a factor of t (1) ; a contradiction.…”

Section: Fine Wordsmentioning

confidence: 69%

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A characterization of fine words over a finite alphabet

Glen

2008

Theoretical Computer Science

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To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w is either a "strict episturmian word" or a strict "skew episturmian word''. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue of Theoretical Computer Science

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Section: Fine Wordsmentioning

confidence: 69%

“…Now we show that F(as (1) ) ⊆ F(t (1) ). Suppose not, i.e., suppose F(as (1) ) ⊆ F(t (1) ). Then there exists a minimal m ∈ N + such that as…”

Section: Fine Wordsmentioning

confidence: 97%

A characterization of fine words over a finite alphabet

Glen

2008

Theoretical Computer Science

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“…Actually, this result was known much earlier, dating back to the work of P. Veerman [112,113] in the mid 80's. Since that time, these 'Sturmian inequalities' have been rediscovered numerous times under different guises, as discussed in the forthcoming survey paper [6]. Continuing his work in relation to inequality (7.1), Pirillo [94] proved further that, in the case of an arbitrary finite alphabet A, an infinite word s ∈ A ω is epistandard if and only if, for any lexicographic order, we have as ≤ min(s) where a = min(A).…”

Section: Extremal Propertiesmentioning

confidence: 99%

Episturmian words: a survey

Glen

Justin

2009

RAIRO-Theor. Inf. Appl.

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In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.

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“…Allouche and Bacher [2] use Toeplitz transformations to construct S n in essentially the same way. Dekking et al [8] have a similar result.…”

Section: Definition 1 (Interleave Operator)mentioning

confidence: 99%

“…Allouche and Bacher [2] generate S using a particular Toeplitz transform. They commence with the sequence…”

Section: The General Alternating Paperfolding Sequencementioning

confidence: 99%

Mirroring and interleaving in the paperfolding sequence

Bates

Bunder

Tognetti

2010

APPL ANAL DISCRETE M

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Three equivalent methods of generating the paperfolding sequence are presented as well as a categorisation of runs of identical terms. We find all repeated subsequences, the largest repeated subsequences and the spacing of singles, doubles and triples throughout the sequence. The paperfolding sequence is shown to have links to the Binary Reflected Gray Code and the Stern-Brocot tree. Three equivalent methods of generating the paperfolding sequence are presented as well as a categorisation of runs of identical terms. We find all repeated subsequences, the largest repeated subsequences and the spacing of singles, doubles and triples throughout the sequence. The paperfolding sequence is shown to have links to the Binary Reflected Gray Code and the Stern-Brocot tree. Disciplines Physical Sciences and Mathematics

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Extremal properties of (epi)Sturmian sequences and distribution modulo $1$

Cited by 10 publications

References 71 publications

A characterization of fine words over a finite alphabet

A characterization of fine words over a finite alphabet

Episturmian words: a survey

Mirroring and interleaving in the paperfolding sequence

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