Abstract
Answering a question of Richomme, Currie and Rampersad proved that 7/3 is the infimum of the real numbers α > 2 such that there exists an infinite binary word that avoids α-powers but is highly 2-repetitive, i.e., contains arbitrarily large squares beginning at every position. In this paper, we prove similar statements about β-repetitive words, for some other β’s, on the binary and the ternary alphabets.
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References
Berstel, J.: Axel Thue’s papers on repetitions in words: a translation. In: Publications du LACIM, vol. 20 (1994)
Brandenburg, F.J.: Uniformly growing k-th powerfree homomorphisms. Theoret. Comput. Sci. 23, 69–82 (1983)
Carpi, A.: On Dejean’s conjecture over large alphabets. Theor. Comput. Sci. 385, 137–151 (2007)
Currie, J., Rampersad, N.: For each α > 2 there is an infinite binary word with critical exponent α. Electron. J. Combinatorics 15, #N34 (2008)
Currie, J., Rampersad, N.: Dejean’s conjecture holds for n ≥ 30. Theoret. Comput. Sci. 410, 2885–2888 (2009)
Currie, J., Rampersad, N.: Infinite words containing squares at every position. Theor. Inform. Appl. 44, 113–124 (2010)
Currie, J., Rampersad, N., Shallit, J.: Binary words containing infinitely many overlaps. Electron. J. Combinatorics 13, #R82 (2006)
Dejean, F.: Sur un théoréme de Thue. J. Combin. Theory Ser. A 13 (1972)
Dekking, F.M.: On repetitions in binary sequences. J. Comb. Theory Ser. A 20, 292–299 (1976)
Krieger, D., Shallit, J.: Every real number greater than 1 is a critical exponent. Theoret. Comput. Sci. 381 (2007)
Mignosi, F., Pirillo, G.: Repetitions in the Fibonacci infinite word. RAIRO Inform. Theor. Appl. 26 (1992)
Pansiot, J.-J.: A propos d’une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7 (1984)
Richomme, G.: Personal communication (2005)
Saari, K.: Everywhere α-repetitive sequences and Sturmian words. Europ. J. Combin. 31, 177–192 (2010)
Shur, A.M.: The structure of the set of cube-free ℤ-words in a two-letter alphabet. Izv. Ross. Akad. Nauk Ser. Mat. 64, 201–224 (2000); English translation in Izv. Math. 64, 847–871 (2000)
Thue, A.: Uber unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana (7) (1906)
Thue, A.: Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana (10) (1912)
Vaslet, E.: Critical exponents of words over 3 letters (submitted)
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Rampersad, N., Vaslet, E. (2011). On Highly Repetitive and Power Free Words. In: Mauri, G., Leporati, A. (eds) Developments in Language Theory. DLT 2011. Lecture Notes in Computer Science, vol 6795. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22321-1_38
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DOI: https://doi.org/10.1007/978-3-642-22321-1_38
Publisher Name: Springer, Berlin, Heidelberg
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