## Dejean’s conjecture holds for n ≥ 27 (2009)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Currie09dejean’sconjecture,

author = {James Currie and Narad Rampersad},

title = {Dejean’s conjecture holds for n ≥ 27},

year = {2009}

}

### OpenURL

### Abstract

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 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 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 n-letter 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). Moulin-Ollagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and Mohammad-Noori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.

### Citations

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Citation Context ...rrie@uwinnipeg.ca n.rampersad@uwinnipeg.ca July 13, 2009 Abstract 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 r-power is a non-empty word w = xx ′ such that x ′ is the pref... |

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Citation Context ...rrie@uwinnipeg.ca n.rampersad@uwinnipeg.ca July 13, 2009 Abstract 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 r-power is a non-empty word w = xx ′ such that x ′ is the pref... |

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Citation Context ...onjecture 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 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... |

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Citation Context ...onjecture 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 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... |

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Citation Context ...onjecture 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 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... |

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Citation Context ...lled the repetitive threshold of an n-letter 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). Moulin-Ollagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14. ∗ The author is... |

11 |
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Citation Context ...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). Moulin-Ollagnier =-=[11]-=- verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14. ∗ The author is supported by an NSERC Discovery Grant. † The author is supported ... |

9 |
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9 |
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Citation Context ... ̸= 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). Moulin-Ollagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie =-=[10]-=- proved the conjecture for 12 ≤ n ≤ 14. ∗ The author is supported by an NSERC Discovery Grant. † The author is supported by an NSERC Postdoctoral Fellowship. 1Recently, Carpi [3] showed that Dejean’s... |

2 | Dejean’s conjecture holds for n ≥ 30
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(Show Context)
Citation Context ...l Fellowship. 1Recently, 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 ≥ 30. (See =-=[4]-=-.) In the present note we show that in fact Dejean’s conjecture holds for n ≥ 27. The following definitions are from [3]: For any non-negative integer r let Ar = {1, 2, . . ., r}. Fix n ≥ 27. Let m = ... |

2 |
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1 | Dejean’s conjecture holds for n ≥ 27, http://arxiv.org/PS cache/arxiv/pdf/0806/0806.0043v2.pdf - Currie, Rampersad |

1 |
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(Show Context)
Citation Context ...Moulin Ollagnier becomes computationally feasible; in a recent paper the present authors proved Dejean’s conjecture by resolving computationally the cases n ≤ 26. Dejean’s conjecture is correct! (See =-=[5]-=-.) The following definitions are from [3]: For any non-negative integer r let Ar = {1, 2, . . ., r}. Fix n ≥ 27. Let m = ⌊(n − 3)/6⌋. Let ker ψ = {v ∈ A∗ m | ∀a ∈ Am, 4 divides |v|a}. (We use this as ... |

1 |
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(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 Mohammad-Noori and Currie =-=[11]-=- proved the conjecture for 12 ≤ n ≤ 14. Recently, Carpi [3] showed that Dejean’s conjecture holds for n ≥ 33. Carpi’s result is computation-free, and resolving Dejean’s conjecture is thus reduced to f... |