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32
On Maximal Repetitions in Words
 J. Discrete Algorithms
, 1999
"... A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. ..."
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A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first study maximal repetitions in Fibonacci words  we count their exact number, and estimate the sum of their exponents. These quantities turn out to be linearlybounded in the length of the word. We then prove that the maximal number of maximal repetitions in general words (on arbitrary alphabet) of length n is linearlybounded in n, and we mention some applications and consequences of this result.
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.
Polynomial versus exponential growth in repetitionfree binary words
 To appear, J. Combinatorial Theory Ser. A
, 2003
"... It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of le ..."
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Cited by 18 (4 self)
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It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of length n that avoid 7 3powers, but there are exponentially many binary words of length n that avoid 7+ 3powers. This answers an open question of Kobayashi from 1986. 1
REVERSALS AND PALINDROMES IN CONTINUED FRACTIONS
"... Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1. ..."
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Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1.
On RepetitionFree Binary Words of Minimal Density
 Theoretical Computer Science
, 1999
"... We study the minimal proportion (density) of one letter in nth powerfree binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for nth powerfree binary w ..."
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Cited by 8 (2 self)
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We study the minimal proportion (density) of one letter in nth powerfree binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for nth powerfree binary words the density function is 1 n + 1 n 3 + 1 n 4 + O( 1 n 5 ). We also consider a generalization of nth powerfree words for fractional powers (exponents): a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in xth powerfree binary words as a function of x and prove, in particular, that this function is discontinuous at 7 3 as well as at all integer points n 3. Finally, we give an estimate of the size of the jumps. Keywords: Unavoidable patterns, powerfree words, exponent, minimal density. 1 Introduction One of classical topics of formal language theory and word combinatorics is th...
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 = 011010011001011010010110 ..."
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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
Proof of Dejean's conjecture for alphabets with 5, 6, 7, 8, 9, 10 and 11 letters
, 1990
"... . Axel Thue proved that overlapping factors could be avoided in arbitrarily long words on a twoletter alphabet while, on the same alphabet, square factors always occur in words longer than 3. Fran¸coise Dejean stated an analogous result for threeletter alphabets : every long enough word has a fact ..."
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. Axel Thue proved that overlapping factors could be avoided in arbitrarily long words on a twoletter alphabet while, on the same alphabet, square factors always occur in words longer than 3. Fran¸coise Dejean stated an analogous result for threeletter alphabets : every long enough word has a factor which is a fractional power with an exponent at least 7=4 and there exist arbitrary long words in which no factor is a fractional power with an exponent strictly greater than 7=4. The number 7=4 is called the repetition threshold of the threeletter alphabets. Thereafter, she proposed the following conjecture : the repetition threshold of the kletter alphabets is equal to k=(k \Gamma 1) except in the particular cases k = 3, where this threshold is 7=4, and k = 4, where it is 7=5. For k = 4, this conjecture was proved by Jean Jacques Pansiot. In this paper, we give a computeraided proof of Dejean's conjecture for several other values : 5; 6; 7; 8; 9; 10; 11. R'esum'e. Axel Thue a montr'...