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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.
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.
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'...
Dejean’s conjecture holds for n ≥ 30
"... We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30. The following definitions are from sections 8 and 9 of [1]: Fix n ≥ 30. Let m = ⌊(n − 3)/6⌋. Let Am = {1, 2,..., m}. Let ker ψ = {v ∈ A ∗ m∀a ∈ Am, 4 divides va}. (In fact, this is not Carpi’s definition of ker ψ, but ..."
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We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30. The following definitions are from sections 8 and 9 of [1]: Fix n ≥ 30. Let m = ⌊(n − 3)/6⌋. Let Am = {1, 2,..., m}. Let ker ψ = {v ∈ A ∗ m∀a ∈ Am, 4 divides va}. (In fact, this is not Carpi’s definition of ker ψ, but rather the assertion of his Lemma 9.1.) A word v ∈ A + m is a ψkernel repetition if it has period q and a prefix v ′ of length q such that v ′ ∈ ker ψ, (n−1)(v+1) ≥ nq − 3. It will be convenient to have the following new definition: If v has period q and its prefix v ′ of length q is in ker ψ, we say that q is a kernel period of v. As Carpi states at the beginning of section 9 of [1]: By the results of the previous sections, at least in the case n ≥ 30, in order to construct an infinite word on n letters avoiding
A Generalization of Repetition Threshold
, 2005
"... Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number α such that there exists an infinite word over a kletter alphabet that avoids βpowers for all β > α. We generalize this concept to include the lengths of the avoided words. We give some ..."
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Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number α such that there exists an infinite word over a kletter alphabet that avoids βpowers for all β > α. We generalize this concept to include the lengths of the avoided words. We give some conjectures supported by numerical evidence and prove some of these conjectures. As a consequence of one of our results, we show that the pattern ABCBABC is 2avoidable. This resolves a question left open in Cassaigne’s thesis.
Dejean’s conjecture holds for n ≥ 27
, 2009
"... We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional rpower is a non ..."
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We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional rpower is a nonempty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)x. For example, 010 is a 3/2power. A basic problem is that of identifying the repetitive threshold for each alphabet size n> 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The infimum is called the repetitive threshold of an nletter alphabet, denoted by RT(n). Dejean’s conjecture [5] is that ⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 n/(n − 1) n ̸ = 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). MoulinOllagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.
Efficient Lower Bounds on the Number of Repetitionfree Words
"... We propose a new effective method for obtaining lower bounds on the number of repetitionfree words over a finite alphabet. 1 ..."
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We propose a new effective method for obtaining lower bounds on the number of repetitionfree words over a finite alphabet. 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