## A PROOF OF DEJEAN’S CONJECTURE (905)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Currie905aproof,

author = {James Currie and Narad Rampersad},

title = {A PROOF OF DEJEAN’S CONJECTURE},

year = {905}

}

### OpenURL

### Abstract

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.

### Citations

695 |
Combinatorics on words
- Lothaire
- 1997
(Show Context)
Citation Context ...jecture holds for n ≥ 27 [4, 5]. In this note we show that in fact Dejean’s conjecture holds for n ≥ 2. We will freely assume the usual notions of combinatorics on words as set forth in, for example, =-=[9]-=-. 2. Morphisms Given previous work, it remains only to show that Dejean’s conjecture holds for 15 ≤ n ≤ 26. This follows from the fact that the following morphisms are ‘convenient’ in the sense of [13... |

127 |
die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, nvss 1
- Thue
- 1912
(Show Context)
Citation Context ... we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1. Introduction Repetitions in words have been studied since the beginning of the previous century =-=[14, 15]-=-. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 6, 7, 8, 10]. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the pre... |

111 |
Über unendliche Zeichenreihen. Norske
- Thue
- 1906
(Show Context)
Citation Context ... we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1. Introduction Repetitions in words have been studied since the beginning of the previous century =-=[14, 15]-=-. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 6, 7, 8, 10]. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the pre... |

41 |
Sur un théorème de Thue
- Dejean
- 1972
(Show Context)
Citation Context ...5 ≤ n ≤ 26. 1. Introduction Repetitions in words have been studied since the beginning of the previous century [14, 15]. Recently, there has been much interest in repetitions with fractional exponent =-=[1, 3, 6, 7, 8, 10]-=-. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifyi... |

36 |
Uniformly growing k-th power-free homomorphisms, Theoret
- Brandenburg
- 1983
(Show Context)
Citation Context ...5 ≤ n ≤ 26. 1. Introduction Repetitions in words have been studied since the beginning of the previous century [14, 15]. Recently, there has been much interest in repetitions with fractional exponent =-=[1, 3, 6, 7, 8, 10]-=-. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifyi... |

31 |
Repetitions in the Fibonacci infinite word
- Mignosi, Pirillo
- 1992
(Show Context)
Citation Context ...5 ≤ n ≤ 26. 1. Introduction Repetitions in words have been studied since the beginning of the previous century [14, 15]. Recently, there has been much interest in repetitions with fractional exponent =-=[1, 3, 6, 7, 8, 10]-=-. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifyi... |

20 | Nonrepetitive sequences on three symbols, Quart - Brinkhuis - 1983 |

17 |
A propos d’une conjecture de F. Dejean sur les répétitions dans les mots
- Pansiot
- 1984
(Show Context)
Citation Context ... the repetitive threshold of an n-letter alphabet, denoted by RT(n). Dejean’s conjecture [6] is that ⎧ ⎪⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 ⎪⎩ n/(n − 1), n ̸= 3, 4. Thue, Dejean and Pansiot, respectively =-=[15, 6, 13]-=-, established the values RT(2), RT(3), RT(4). Moulin Ollagnier [12] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [11] proved the conjecture for 12 ≤ n ≤ 14. Recently, Carp... |

12 |
On Dejean’s conjecture over large alphabets, Theoret
- Carpi
(Show Context)
Citation Context |

10 |
On critical exponents in fixed points of non-erasing morphisms
- Krieger
- 2006
(Show Context)
Citation Context |

10 |
Proof of Dejean’s conjecture for alphabets with 5,6,7,8,9,10 and 11 letters, Theoret
- Moulin-Ollagnier
- 1992
(Show Context)
Citation Context ...an’s conjecture [6] is that ⎧ ⎪⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 ⎪⎩ n/(n − 1), n ̸= 3, 4. Thue, Dejean and Pansiot, respectively [15, 6, 13] established the values RT(2), RT(3), RT(4). Moulin-Ollagnier =-=[12]-=- verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [11] proved the conjecture for 12 ≤ n ≤ 14. Recently, Carpi [3] showed that Dejean’s conjecture holds for n ≥ 33. The present... |

9 |
Dejean’s conjecture and Sturmian words
- Mohammad-Noori, Currie
(Show Context)
Citation Context ...= 3, 4. Thue, Dejean and Pansiot, respectively [15, 6, 13], established the values RT(2), RT(3), RT(4). Moulin Ollagnier [12] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie =-=[11]-=- proved the conjecture for 12 ≤ n ≤ 14. Recently, Carpi [3] showed that Dejean’s conjecture holds for n ≥ 33. The present authors improved one of Carpi’s constructions to show that Dejean’s conjecture... |

3 |
On the growth rates of complexity of threshold languages
- Shur, Gorbunova
- 2008
(Show Context)
Citation Context ...g all binary words of length at most r which are Pansiot encodings of ( ) n +-free words over Σn. Initially this was part of our n−1 strategy. Unfortunately, our experience supports the conjecture in =-=[15]-=-, that the number of these words grows approximately as 1.24r (independently of n.) For successive r values we looked at all possible pairs 〈h(0), h(1)〉 such that |h(0)|, |h(1)| ≤ r where h(0), h(1) w... |

2 | Dejean’s conjecture holds for n ≥ 30
- Currie, Rampersad
(Show Context)
Citation Context ...ture for 12 ≤ n ≤ 14. Recently, Carpi [3] showed that Dejean’s conjecture holds for n ≥ 33. The present authors improved one of Carpi’s constructions to show that Dejean’s conjecture holds for n ≥ 27 =-=[4, 5]-=-. In this note we show that in fact Dejean’s conjecture holds for n ≥ 2. 2. Results In this preliminary version we list for each n a “convenient” (in the sense of Moulin Ollagnier [12]) morphism that ... |

2 |
A generalization of repetition threshold, Theoret
- Ilie, Ochem, et al.
(Show Context)
Citation Context |

2 |
Ollagnier, Proof of Dejean’s conjecture for alphabets with 5
- Moulin
- 1992
(Show Context)
Citation Context ...n’s conjecture [6] is that ⎧ ⎪⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 ⎪⎩ n/(n − 1), n ̸= 3, 4. Thue, Dejean and Pansiot, respectively [15, 6, 13], established the values RT(2), RT(3), RT(4). Moulin Ollagnier =-=[12]-=- verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [11] proved the conjecture for 12 ≤ n ≤ 14. Recently, Carpi [3] showed that Dejean’s conjecture holds for n ≥ 33. The present... |

1 | Dejean’s conjecture holds for n ≥ 27
- Currie, Rampersad
(Show Context)
Citation Context ...ture for 12 ≤ n ≤ 14. Recently, Carpi [3] showed that Dejean’s conjecture holds for n ≥ 33. The present authors improved one of Carpi’s constructions to show that Dejean’s conjecture holds for n ≥ 27 =-=[4, 5]-=-. In this note we show that in fact Dejean’s conjecture holds for n ≥ 2. 2. Results In this preliminary version we list for each n a “convenient” (in the sense of Moulin Ollagnier [12]) morphism that ... |