## Efficient Lower Bounds on the Number of Repetition-free Words

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@MISC{Kolpakov_efficientlower,

author = {Roman Kolpakov},

title = {Efficient Lower Bounds on the Number of Repetition-free Words},

year = {}

}

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### Abstract

We propose a new effective method for obtaining lower bounds on the number of repetition-free words over a finite alphabet. 1

### Citations

126 |
Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihern. Norske vid
- Thue
- 1912
(Show Context)
Citation Context ...-free (cube-free) if it contains no squares (cubes) as factors. It is easy to see that there are no binary square-free words of length more than 3. On the other hand, by the classical results of Thue =-=[19, 20]-=-, there exist ternary square-free words of arbitrary length and binary cube-free words of arbitrary length. If we denote by S 〈sf〉 (n) the number of ternary square-free words of length n and by S 〈cf〉... |

107 |
Uber unendliche zeichenreihen. Norske Vid
- Thue
- 1906
(Show Context)
Citation Context ...-free (cube-free) if it contains no squares (cubes) as factors. It is easy to see that there are no binary square-free words of length more than 3. On the other hand, by the classical results of Thue =-=[19, 20]-=-, there exist ternary square-free words of arbitrary length and binary cube-free words of arbitrary length. If we denote by S 〈sf〉 (n) the number of ternary square-free words of length n and by S 〈cf〉... |

41 |
Sur un théorème de Thue
- Dejean
- 1972
(Show Context)
Citation Context ... by S 〈cf〉 (n) the number of binary cube-free words of length n, we then have that S 〈sf〉 (n) > 0 and S 〈cf〉 (n) > 0 for any n. For ternary square-free words this result was strengthened by Dejean in =-=[7]-=-. She found ternary words of arbitrary length which have no factors with exponents greater than 7/4. On the other hand, she showed that any long enough ternary word contains a factor with an exponent ... |

36 |
Uniformly growing k-th power-free homomorphisms, Theoret
- Brandenburg
- 1983
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Citation Context ...ary words are factorial. So there exist the growth rates γ 〈sf〉 � n = limn→∞ S 〈sf〉 (n), γ 〈cf〉 � n = limn→∞ S 〈cf〉 (n), γ 〈lf〉 � n = limn→∞ S 〈lf〉 (n) of words from these sets. Brandenburg proved in =-=[3]-=- that the values S 〈sf〉 (n) and S 〈cf〉 (n) grew exponentially with n, namely, S 〈sf〉 (n) ≥ 6 · 1.032 n and S 〈cf〉 (n) ≥ 2 · 1.080 n , i. e. γ 〈sf〉 ≥ 1.032 and γ 〈cf〉 ≥ 1.080. Later the lower bound for... |

26 | Open problems in pattern avoidance - Currie - 1993 |

20 |
Nonrepetitive sequences on three symbols, Quart
- Brinkhuis
- 1983
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Citation Context ...f for any word w from W all factors of w are also contained in W. We denote by W(n) the subset of W consisting of all words of length n. If W is factorial then it is not difficult to show (see, e.g., =-=[4, 1]-=-) that there exists the limit 1 This work is supported by the program of the President of the Russian Federation for supporting of young researchers and scientific schools (Grants MD–3635.2005.1 and N... |

17 |
A propos d’une conjecture de F. Dejean sur les répétitions dans les mots
- Pansiot
- 1984
(Show Context)
Citation Context ...he minimal limit for exponents of prohibited factors in arbitrarily long words over a k-letter alphabet is equal to 7/5 for k = 4 and k/k −1 for k ≥ 5. This conjecture was proved for k = 4 by Pansiot =-=[17]-=-, for 5 ≤ k ≤ 11 by Moulin Ollagnier [14], for 12 ≤ k ≤ 14 by Mohammad-Noori and Currie [13], and for k ≥ 38 by Carpi [5]. Denote by S 〈lf〉 (n) the number of minimally repetitive ternary words of leng... |

16 | There are more than 2n/17 n-letter ternary square-free words, J. Integer Seq. 1: Article 98.1.9
- Ekhad, Zeilberger
- 1998
(Show Context)
Citation Context ... S 〈sf〉 (n) ≥ 6 · 1.032 n and S 〈cf〉 (n) ≥ 2 · 1.080 n , i. e. γ 〈sf〉 ≥ 1.032 and γ 〈cf〉 ≥ 1.080. Later the lower bound for γ 〈sf〉 was improved consecutively 3 by Ekhad, Zeilberger, Grimm, and Sun in =-=[9, 10, 18]-=-. The best upper bounds known at present γ 〈sf〉 < 1.30178858 and γ 〈cf〉 < 1.4576 are obtained by Ochem and Edlin in [16] and [8] respectively. In [15] Ochem established the exponential growth of the n... |

14 | Improved bounds on the number of ternary square-free words, J. Integer Seq. 4: Article 01.2.7
- Grimm
- 2001
(Show Context)
Citation Context ... S 〈sf〉 (n) ≥ 6 · 1.032 n and S 〈cf〉 (n) ≥ 2 · 1.080 n , i. e. γ 〈sf〉 ≥ 1.032 and γ 〈cf〉 ≥ 1.080. Later the lower bound for γ 〈sf〉 was improved consecutively 3 by Ekhad, Zeilberger, Grimm, and Sun in =-=[9, 10, 18]-=-. The best upper bounds known at present γ 〈sf〉 < 1.30178858 and γ 〈cf〉 < 1.4576 are obtained by Ochem and Edlin in [16] and [8] respectively. In [15] Ochem established the exponential growth of the n... |

11 | The entropy of squarefree words
- Baake, Elaser, et al.
- 1997
(Show Context)
Citation Context ...f for any word w from W all factors of w are also contained in W. We denote by W(n) the subset of W consisting of all words of length n. If W is factorial then it is not difficult to show (see, e.g., =-=[4, 1]-=-) that there exists the limit 1 This work is supported by the program of the President of the Russian Federation for supporting of young researchers and scientific schools (Grants MD–3635.2005.1 and N... |

9 |
Dejean’s conjecture and Sturmian words
- Mohammad-Noori, Currie
(Show Context)
Citation Context ...er alphabet is equal to 7/5 for k = 4 and k/k −1 for k ≥ 5. This conjecture was proved for k = 4 by Pansiot [17], for 5 ≤ k ≤ 11 by Moulin Ollagnier [14], for 12 ≤ k ≤ 14 by Mohammad-Noori and Currie =-=[13]-=-, and for k ≥ 38 by Carpi [5]. Denote by S 〈lf〉 (n) the number of minimally repetitive ternary words of length n. It follows from the result of Dejean that S 〈lf〉 (n) > 0 for any n. Note that the set ... |

7 | New lower bound on the number of ternary square free words, preprint (2002); see also http://www.math.temple.edu/~xysun
- Sun
(Show Context)
Citation Context ... S 〈sf〉 (n) ≥ 6 · 1.032 n and S 〈cf〉 (n) ≥ 2 · 1.080 n , i. e. γ 〈sf〉 ≥ 1.032 and γ 〈cf〉 ≥ 1.080. Later the lower bound for γ 〈sf〉 was improved consecutively 3 by Ekhad, Zeilberger, Grimm, and Sun in =-=[9, 10, 18]-=-. The best upper bounds known at present γ 〈sf〉 < 1.30178858 and γ 〈cf〉 < 1.4576 are obtained by Ochem and Edlin in [16] and [8] respectively. In [15] Ochem established the exponential growth of the n... |

5 |
Upper bound on the number of ternary square-free words
- Ochem, Reix
- 2006
(Show Context)
Citation Context ...sf〉 was improved consecutively 3 by Ekhad, Zeilberger, Grimm, and Sun in [9, 10, 18]. The best upper bounds known at present γ 〈sf〉 < 1.30178858 and γ 〈cf〉 < 1.4576 are obtained by Ochem and Edlin in =-=[16]-=- and [8] respectively. In [15] Ochem established the exponential growth of the number of minimally repetitive words over either a three-letter or a four-letter alphabet. However, this result does not ... |

3 |
The number of binary cube-free words of length up to 47 and their numerical analysis
- Edlin
- 1999
(Show Context)
Citation Context ...mproved consecutively 3 by Ekhad, Zeilberger, Grimm, and Sun in [9, 10, 18]. The best upper bounds known at present γ 〈sf〉 < 1.30178858 and γ 〈cf〉 < 1.4576 are obtained by Ochem and Edlin in [16] and =-=[8]-=- respectively. In [15] Ochem established the exponential growth of the number of minimally repetitive words over either a three-letter or a four-letter alphabet. However, this result does not give any... |

2 | Growth of repetition-free words – a review. Theoret - Berstel |

2 |
On the repetition threshold for large alphabets
- Carpi
- 2006
(Show Context)
Citation Context ...r k = 4 and k/k −1 for k ≥ 5. This conjecture was proved for k = 4 by Pansiot [17], for 5 ≤ k ≤ 11 by Moulin Ollagnier [14], for 12 ≤ k ≤ 14 by Mohammad-Noori and Currie [13], and for k ≥ 38 by Carpi =-=[5]-=-. Denote by S 〈lf〉 (n) the number of minimally repetitive ternary words of length n. It follows from the result of Dejean that S 〈lf〉 (n) > 0 for any n. Note that the set of all ternary square-free wo... |

2 |
A generalization of repetition threshold, Theoret
- Ilie, Ochem, et al.
(Show Context)
Citation Context ...provided that the growth is exponential). Moreover, this method can be easily modified for the case when additional restrictions are imposed on the minimal value of periods of prohibited factors (see =-=[11]-=-). We suppose that this method can be also generalized for estimation of growth rates of words avoiding patterns (see, e.g., [6]). Acknowledgments The author is grateful to the referee of this paper f... |

2 |
On the number of repetition-free words
- Kolpakov
- 2006
(Show Context)
Citation Context ...the exponential growth of the number of minimally repetitive words over either a three-letter or a four-letter alphabet. However, this result does not give any significant lower bound for γ 〈lf〉 . In =-=[12]-=- we proposed a new method for obtaining lower bounds on the number of repetitionfree words. This method is essentially based on inductive estimation of the number of words which contain repetitions as... |

2 |
Ollagnier, Proof of Dejean’s conjecture for alphabets with 5
- Moulin
- 1992
(Show Context)
Citation Context ...ted factors in arbitrarily long words over a k-letter alphabet is equal to 7/5 for k = 4 and k/k −1 for k ≥ 5. This conjecture was proved for k = 4 by Pansiot [17], for 5 ≤ k ≤ 11 by Moulin Ollagnier =-=[14]-=-, for 12 ≤ k ≤ 14 by Mohammad-Noori and Currie [13], and for k ≥ 38 by Carpi [5]. Denote by S 〈lf〉 (n) the number of minimally repetitive ternary words of length n. It follows from the result of Dejea... |

2 |
A generator of morphisms for infinite words
- Ochem
- 2004
(Show Context)
Citation Context ... 3 by Ekhad, Zeilberger, Grimm, and Sun in [9, 10, 18]. The best upper bounds known at present γ 〈sf〉 < 1.30178858 and γ 〈cf〉 < 1.4576 are obtained by Ochem and Edlin in [16] and [8] respectively. In =-=[15]-=- Ochem established the exponential growth of the number of minimally repetitive words over either a three-letter or a four-letter alphabet. However, this result does not give any significant lower bou... |