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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
There Are Ternary Circular SquareFree Words of Length n for n ≥ 18
, 2002
"... There are circular squarefree words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson. ..."
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There are circular squarefree words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson.
Binary words containing infinitely many overlaps
"... We characterize the squares occurring in infinite overlapfree binary words and construct various α powerfree binary words containing infinitely many overlaps. 1 ..."
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We characterize the squares occurring in infinite overlapfree binary words and construct various α powerfree binary words containing infinitely many overlaps. 1
A rewriting of Fife’s theorem about overlapfree words
 Results and Trends in Theoretical Computer Science, LNCS 812
"... The purpose of this expository paper is to present a selfcontained proof of a famous theorem of Fife that gives a full description of the set of infinite overlapfree words over a binary alphabet. Fife's characterization consists in a parameterization of these infinite words by a set of infini ..."
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The purpose of this expository paper is to present a selfcontained proof of a famous theorem of Fife that gives a full description of the set of infinite overlapfree words over a binary alphabet. Fife's characterization consists in a parameterization of these infinite words by a set of infinite words over a ternary alphabet. The result is that the latter is a regular set. The proof is by the explicit construction of the minimal automaton, obtained by the method of left quotients.
There Exist Binary Circular 5/2+ Power Free Words of Every Length
, 2004
"... We show that there exist binary circular 5=2 power free words of every length. ..."
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We show that there exist binary circular 5=2 power free words of every length.
OverlapFree Symmetric D0L words
, 2001
"... Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 0110100110010110100 ..."
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Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 01101001100101101001011001101001 : : :: It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlapfree words was studied e. g. by Fife [8] who described all binary overlapfree infinite words and Seebold [13] who proved that the ThueMorse word is essentially the only binary overlapfree word which is a fixed point of a morphism. Nowadays the theory of overlapfree words is a part of a more general theory of pattern avoidance [5]. J.P. Allouche and J. Shallit [2] asked if the initial Thue's construction of an overlapfree wo
A characterization of 2+free words over a binary alphabet
, 1995
"... It is shown that 2+repetition, i.e. a word of the form uvuvu where u is a letter and v is a word, is the smallest repetition which can be avoided in infinite words over binary alphabet. Such binary words avoiding pattern uvuvu, finite or infinite, are called as 2+free words and those words are the ..."
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It is shown that 2+repetition, i.e. a word of the form uvuvu where u is a letter and v is a word, is the smallest repetition which can be avoided in infinite words over binary alphabet. Such binary words avoiding pattern uvuvu, finite or infinite, are called as 2+free words and those words are the main topic of this work. It is shown here that 2+free words over binary alphabet can be presented as words built from special kind of blocks, called Morseblocks, with some rules. In particular, the given presentation by these blocks is unique for 2+free words long enough. Moreover, it is also shown that the language generated by this presentation can be described by some automaton. In fact, the corresponding presentation in blocks for finite 2
Infinite words containing squares at every position
 In Proceedings of Journées Montoises D’Informatique Théorique
, 2008
"... Richomme asked the following question: what is the infimum of the real numbers α> 2 such that there exists an infinite word that avoids αpowers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer ..."
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Richomme asked the following question: what is the infimum of the real numbers α> 2 such that there exists an infinite word that avoids αpowers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is α = 7/3. 1
Words strongly avoiding fractional powers
 Europ. J. of Combinatorics
, 1999
"... Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be writt ..."
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Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be written xyyz, where x; y; z are words, and y is nonempty. Infinite