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 $\{1,2,\ldots, k\}$. Shur studied the asymptotic behavior of $\alpha(k,\beta)$ for $\beta\ge2$ 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 $\frac{9}{8}\le\beta<2$ the asymptotic upper-bound holds of his conjecture holds. We show that the asymptotic lower-bound of his conjecture holds. This implies that the conjecture is true for $\frac{9}{8}\le\beta<2$.

翻译：我们研究一些无功率语言的增长率。 对于任何整数美元和实值$\beta>1美元的增长率, 我们让$\alpha( k,\beta) 美元作为字母1, 2,\ldots, k ⁇ 。 Shur 研究了$\alpha( k,\beta) $( beta) 的无药性行为。 对于任何整数美元和实值$\beta>1美元的增长率, 我们让$\alpha( k,\beta) 美元成为字母1\ beta < 2$的无药性字数的增长率。 Shur 研究了 $\ alpha( k,\beta) $( beta) 和 $( beta) 的无药性上限值为$2美元。 我们显示, 其直觉测值的无药性下限值为 。 这意味着对 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\