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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
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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Cited by 2 (1 self)
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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
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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Cited by 1 (0 self)
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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
How Many Square Occurrences Must a Binary Sequence Contain?
, 2003
"... Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sucient to construct an in nite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We s ..."
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Cited by 1 (0 self)
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Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sucient to construct an in nite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We show that this quantity is, in the limit, a constant fraction of the word length, and prove that this constant is 0:55080:::.
On the Entropy and Letter Frequencies of Powerfree Words
, 811
"... We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of bina ..."
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We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80. 1
Theoretical Informatics and Applications Informatique Théorique et Applications Will be set by the publisher DEJEAN’S CONJECTURE AND LETTER FREQUENCY
, 2009
"... Abstract. We prove two cases of a strong version of Dejean’s conjecture involving extremal “ letter frequencies. The results are that there 5 + exist an infinite 4free word over a 5 letter alphabet with letter and an infinitefree word over a 6 letter alphabet frequency 1 6 with letter frequency 1 ..."
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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 5 + exist an infinite 4free word over a 5 letter alphabet with letter and an infinitefree word over a 6 letter alphabet frequency 1 6 with letter frequency 1 5.
Author manuscript, published in "WORDS 2007, France (2007)" Unequal letter frequencies in ternary squarefree words
, 2007
"... We consider the set S of triples (x,y, z) corresponding to the frequency of each alphabet letter in some infinite ternary squarefree word (so x + y +z = 1). We conjecture that this set is convex. We obtain bounds on S by with a generalization of our method to bound the extremal frequency of one let ..."
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We consider the set S of triples (x,y, z) corresponding to the frequency of each alphabet letter in some infinite ternary squarefree word (so x + y +z = 1). We conjecture that this set is convex. We obtain bounds on S by with a generalization of our method to bound the extremal frequency of one letter. This method uses weights on the alphabet letters. Finally, we obtain positive results, that is, explicit triples in S lying close to its boundary. 1 Introduction and preliminary results A square is a repetition of the form xx, where x is a nonempty word; an example in English is hotshots. Let Σk denote the kletter alphabet {0, 1,..., k − 1}. It is easy to see that every word of length ≥ 4 over Σ2 must contain a square, so squares cannot be