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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.
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'...
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.
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 nonemp ..."
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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
Dejean’s conjecture and letter frequency
"... Abstract. We prove two cases of a strong version of Dejean’s conjecture involving “ extremal letter frequencies. The results are that there exist an 5+ infinite 4free word over a 5 letter alphabet with letter frequency and an infinitefree word over a 6 letter alphabet with letter ..."
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Abstract. We prove two cases of a strong version of Dejean’s conjecture involving “ extremal letter frequencies. The results are that there exist an 5+ infinite 4free word over a 5 letter alphabet with letter frequency and an infinitefree word over a 6 letter alphabet with letter
Subwords and Power Free Words Are Not Expressible By Word Equations
"... We consider several open problems of Karhumaki, Mignosi, and Plandowski, cf. [KMP], concerning the expressibility of languages and relations as solutions of word equations. We show first that the (scattered) subword relation is not expressible. Then, we consider the set of kpower free finite words ..."
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We consider several open problems of Karhumaki, Mignosi, and Plandowski, cf. [KMP], concerning the expressibility of languages and relations as solutions of word equations. We show first that the (scattered) subword relation is not expressible. Then, we consider the set of kpower free finite words and solve it negativelly for all nontrivial integer values of k. Finally, we consider the Fibonacci finite words. We do not solve the problem of the expressibility of the set of these words but prove that it cannot be given a negative answer (as believed) using the tools in [KMP]. 1
On ternary squarefree circular words
"... Circular words are cyclically ordered finite sequences of letters. We give a computerfree proof of the following result by Currie: squarefree circular words over the ternary alphabet exist for all lengths l except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ter ..."
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Circular words are cyclically ordered finite sequences of letters. We give a computerfree proof of the following result by Currie: squarefree circular words over the ternary alphabet exist for all lengths l except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ternary squarefree circular words and closed walks in the K3,3 graph. In addition, our proof implies an exponential lower bound on the number of such circular words of length l and allows one to list all lengths l for which such a circular word is unique up to isomorphism.