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 \(\frac {9}{8}\le \beta <2\).

我们研究了一些无功能语言的增长率。对于任何整数k和实数β > 1，我们令α（k，β）为在字母{1,2，…，k }上给定长度的无β词的数量的增长率。舒尔研究的渐近行为α（ķ，β）为β ≥2作为ķ趋向无穷大。他提出了一个关于α（k，β）的渐近行为的猜想，当1 < β时k趋于无穷大。<2。他表明对于\（\ frac {9} {8} \ le \ beta <2 \），渐近上限成立。我们证明了他的猜想的渐近下界成立。这意味着\（\ frac {9} {8} \ le \ beta <2 \）的猜想是正确的。