## Dejean’s conjecture and letter frequency

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@MISC{Chalopin_dejean’sconjecture,

author = {Jérémie Chalopin and Pascal Ochem and Université Bordeaux},

title = {Dejean’s conjecture and letter frequency},

year = {}

}

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### Abstract

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 4-free word over a 5 letter alphabet with letter frequency and an infinite-free word over a 6 letter alphabet with letter

### Citations

107 | Combinatorics of words
- Choffrut, Karhumäki
- 1997
(Show Context)
Citation Context ... [6] proved that Dejean’s conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori [2] also proved the cases 12 ≤ k ≤ 14, and Carpi [3] settled the cases k ≥ 38. For more information, see =-=[1]-=-. In a previous paper, we proposed the following conjecture which implies Dejean’s conjecture. exists an infinite (α + )-free word over Σk. Dejean proved that α(3) = 7 4 also conjectured that α(4) = 7... |

41 |
Sur un théorème de Thue
- Dejean
- 1972
(Show Context)
Citation Context ...frequency in w is n(w) |w| . The repetition threshold is the least exponent α = α(k) such that there . She k and α(k) = k−1 for k ≥ 5. This conjecture is now “almost” solved: Pansiot [9] proved that α=-=(4)-=- = 7 5 and Moulin-Ollagnier [6] proved that Dejean’s conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori [2] also proved the cases 12 ≤ k ≤ 14, and Carpi [3] settled the cases k ≥ 38.... |

18 |
A propos d’une conjecture de F. Dejean sur les répétitions dans les mots
- Pansiot
- 1984
(Show Context)
Citation Context ...w. So the letter frequency in w is n(w) |w| . The repetition threshold is the least exponent α = α(k) such that there . She k and α(k) = k−1 for k ≥ 5. This conjecture is now “almost” solved: Pansiot =-=[9]-=- proved that α(4) = 7 5 and Moulin-Ollagnier [6] proved that Dejean’s conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori [2] also proved the cases 12 ≤ k ≤ 14, and Carpi [3] settled ... |

11 | On Dejean’s conjecture over large alphabets - Carpi |

10 |
Proof of Dejean’s conjecture for alphabets with 5
- Moulin-Ollagnier
- 1992
(Show Context)
Citation Context ...e repetition threshold is the least exponent α = α(k) such that there . She k and α(k) = k−1 for k ≥ 5. This conjecture is now “almost” solved: Pansiot [9] proved that α(4) = 7 5 and Moulin-Ollagnier =-=[6]-=- proved that Dejean’s conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori [2] also proved the cases 12 ≤ k ≤ 14, and Carpi [3] settled the cases k ≥ 38. For more information, see [1].... |

9 |
The minimal density of a letter in an infinite ternary squarefree word is 0.2746
- Tarannikov
- 2002
(Show Context)
Citation Context ... repetitions. Given such a language L, we denote by fmin (resp. fmax) the minimal (resp. maximal) letter frequency in an infinite word that belong to L. Letter frequencies have been mainly studied in =-=[5, 7, 10, 11]-=-. Let Σi denote the i-letter alphabet {0, 1, . . . , i − 1}. We consider here the frequency of the letter 0. Let n(w) denote the number of occurrences of 0 in the finite word w. So the letter frequenc... |

9 | Dejean’s conjecture and Sturmian words - Mohammad-Noori, Currie |

8 | On repetition-free binary words of minimal density, Theoret
- Kolpakov, Kucherov, et al.
- 1999
(Show Context)
Citation Context ... repetitions. Given such a language L, we denote by fmin (resp. fmax) the minimal (resp. maximal) letter frequency in an infinite word that belong to L. Letter frequencies have been mainly studied in =-=[5, 7, 10, 11]-=-. Let Σi denote the i-letter alphabet {0, 1, . . . , i − 1}. We consider here the frequency of the letter 0. Let n(w) denote the number of occurrences of 0 in the finite word w. So the letter frequenc... |

6 |
Letter frequency in infinite repetition-free words, Theoret
- Ochem
(Show Context)
Citation Context ... repetitions. Given such a language L, we denote by fmin (resp. fmax) the minimal (resp. maximal) letter frequency in an infinite word that belong to L. Letter frequencies have been mainly studied in =-=[5, 7, 10, 11]-=-. Let Σi denote the i-letter alphabet {0, 1, . . . , i − 1}. We consider here the frequency of the letter 0. Let n(w) denote the number of occurrences of 0 in the finite word w. So the letter frequenc... |

5 |
Upper bound on the number of ternary square-free words
- Ochem, Reix
- 2006
(Show Context)
Citation Context ...which is probably true). On the other hand, the growth rate of these words is smaller than those of � k k−1 + � -free words. For example, the growth rate of � 5 4 + � -free 5-ary words is about 1.159 =-=[8]-=-, whereas 1.048 is a rough upper bound on the growth rate of � 5 4 + � -free 5-ary words with letter frequency 1 6 .s124 Letter frequency in infinite repetition-free words � 6 + Other cases of Conject... |

5 | On the entropy and letter frequencies of ternary square-free words, Electron
- Richard, Grimm
(Show Context)
Citation Context |

1 |
Dejean’s conjecture and sturmian words, Manuscript
- Currie, Mohammad-Noori
(Show Context)
Citation Context ...or k ≥ 5. This conjecture is now “almost” solved: Pansiot [9] proved that α(4) = 7 5 and Moulin-Ollagnier [6] proved that Dejean’s conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori =-=[2]-=- also proved the cases 12 ≤ k ≤ 14, and Carpi [3] settled the cases k ≥ 38. For more information, see [1]. In a previous paper, we proposed the following conjecture which implies Dejean’s conjecture. ... |

1 |
On the repetition threshold for large alphabets, MFCS
- Carpi
- 2006
(Show Context)
Citation Context ... Pansiot [9] proved that α(4) = 7 5 and Moulin-Ollagnier [6] proved that Dejean’s conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori [2] also proved the cases 12 ≤ k ≤ 14, and Carpi =-=[3]-=- settled the cases k ≥ 38. For more information, see [1]. In a previous paper, we proposed the following conjecture which implies Dejean’s conjecture. exists an infinite (α + )-free word over Σk. Deje... |