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Lower-Bounds on the Growth of Power-Free Languages Over Large Alphabets

Abstract

We study the growth rate of some power-free languages. For any integer k and real β > 1, we let α(k,β) be the growth rate of the number of β-free words of a given length over the alphabet {1,2,…,k}. Shur studied the asymptotic behavior of α(k,β) for β ≥ 2 as k goes to infinity. He suggested a conjecture regarding the asymptotic behavior of α(k,β) as k goes to infinity when 1 < β < 2. He showed that for \(\frac {9}{8}\le \beta <2\) the asymptotic upper-bound holds. We show that the asymptotic lower bound of his conjecture holds. This implies that the conjecture is true for \(\frac {9}{8}\le \beta <2\).

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Correspondence to Matthieu Rosenfeld.

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Rosenfeld, M. Lower-Bounds on the Growth of Power-Free Languages Over Large Alphabets. Theory Comput Syst 65, 1110–1116 (2021). https://doi.org/10.1007/s00224-021-10040-1

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  • DOI: https://doi.org/10.1007/s00224-021-10040-1

Keywords

  • Combinatorics on words
  • Power-free languages
  • Fractional repetitions
  • Exponential growth