Résumé
We study the growth rate of some power-free languages. For any integer k and real beta > 1, we let alpha (k, beta) be the growth rate of the number of beta-free words of a given length over the alphabet {1, 2, ..., k}. Shur studied the asymptotic behavior of alpha (k, beta) for beta >= 2 as k goes to infinity. He suggested a conjecture regarding the asymptotic behavior of alpha (k, beta) as k goes to infinity when 1 < beta < 2. He showed that for 9/8 <= 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 9/8 <= beta < 2.