Dejean's conjecture and letter frequency

Chalopin, Jérémie; Ochem, Pascal

doi:10.1051/ita:2008013

Chalopin, Jérémie ; Ochem, Pascal

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 3, pp. 477-480.

Résumé

We prove two cases of a strong version of Dejean’s conjecture involving extremal letter frequencies. The results are that there exist an infinite $({\frac{5}{4}}^{+})$ -free word over a 5 letter alphabet with letter frequency $\frac{1}{6}$ and an infinite $({\frac{6}{5}}^{+})$ -free word over a 6 letter alphabet with letter frequency $\frac{1}{5}$ .

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/ita:2008013

Classification : 68R15

@article{ITA_2008__42_3_477_0,
     author = {Chalopin, J\'er\'emie and Ochem, Pascal},
     title = {Dejean's conjecture and letter frequency},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {477--480},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {3},
     year = {2008},
     doi = {10.1051/ita:2008013},
     mrnumber = {2434030},
     zbl = {1147.68612},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2008013/}
}

TY  - JOUR
AU  - Chalopin, Jérémie
AU  - Ochem, Pascal
TI  - Dejean's conjecture and letter frequency
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2008
SP  - 477
EP  - 480
VL  - 42
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita:2008013/
DO  - 10.1051/ita:2008013
LA  - en
ID  - ITA_2008__42_3_477_0
ER  -

%0 Journal Article
%A Chalopin, Jérémie
%A Ochem, Pascal
%T Dejean's conjecture and letter frequency
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2008
%P 477-480
%V 42
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita:2008013/
%R 10.1051/ita:2008013
%G en
%F ITA_2008__42_3_477_0

Chalopin, Jérémie; Ochem, Pascal. Dejean's conjecture and letter frequency. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 3, pp. 477-480. doi : 10.1051/ita:2008013. http://archive.numdam.org/articles/10.1051/ita:2008013/

Bibliographie
Cité par

[1] A. Carpi, On Dejeans conjecture over large alphabets. Theoret. Comput. Sci. 385 (2007) 137-151. | MR | Zbl

[2] C. Choffrut and J. Karhumäki, Combinatorics of words, in Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa, Springer-Verlag (1997) 329-438. | MR

[3] M. Mohammad-Noori and J.D. Currie, Dejean's conjecture and Sturmian words. Eur. J. Combin. 28(3) (2007) 876-890. | MR | Zbl

[4] F. Dejean, Sur un théorème de Thue. J. Combin. Theor. Ser. A 13 (1972) 90-99. | MR | Zbl

[5] A. Khalyavin. The minimal density of a letter in an infinite ternary square-free word is $883 / 3215$ . J. Integer Sequences 10 (2007) 07.6.5. | MR | Zbl

[6] R. Kolpakov, G. Kucherov and Y. Tarannikov, On repetition-free binary words of minimal density. Theoret. Comput. Sci. 218 (1999) 161-175. | MR | Zbl

[7] J. Moulin-Ollagnier, Proof of Dejean’s conjecture for alphabets with $5, 6, 7, 8, 9, 10$ and $11$ letters. Theoret. Comput. Sci. 95 (1992) 187-205. | MR | Zbl

[8] P. Ochem, Letter frequency in infinite repetition-free words. Theoret. Comput. Sci. 380 (2007) 388-392. | MR | Zbl

[9] P. Ochem and T. Reix, Upper bound on the number of ternary square-free words, in Workshop on Words and Automata (WOWA'06). St. Petersburg, Russia, June 7 (2006).

[10] J.-J. Pansiot, À propos d'une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7 (1984) 297-311. | MR | Zbl

[11] C. Richard and U. Grimm, On the entropy and letter frequencies of ternary square-free words. Electron. J. Comb. 11 (2004) #R14. | MR | Zbl

[12] Y. Tarannikov, The minimal density of a letter in an infinite ternary square-free word is $0.2746 . . .$ J. Integer Sequences 5 (2002) 02.2.2. | MR | Zbl

Cité par Sources :