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 $\frac {9}{8}\le \beta <2$.