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60
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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Cited by 55 (6 self)
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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.
Polynomial versus exponential growth in repetitionfree words
 J. Comb. Th. A
, 2004
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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.
Highly nonrepetitive sequences: winning strategies from the Local Lemma”. Manuscript available at http://people.cs.uchicago.edu/ ∼wes/seqgame.pdf
"... We prove gametheoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Local Lemma which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the signific ..."
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We prove gametheoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Local Lemma which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the significant dependencies of events. This appears to represent the first successful application of a Local Lemma to games. 1
A PROOF OF DEJEAN’S CONJECTURE
, 905
"... Abstract. We prove Dejean’s conjecture. Specifically, we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1. ..."
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Abstract. We prove Dejean’s conjecture. Specifically, we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 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 powerfre ..."
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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...
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.
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.