We study some properties of the growth rate of $\mathcal{L}(\mathcal{A},\mathcal{F})$, that is, the language of words over the alphabet $\mathcal{A}$ avoiding the set of forbidden factors $\mathcal{F}$. We first provide a sufficient condition on $\mathcal{F}$ and $\mathcal{A}$ for the growth of $\mathcal{L}(\mathcal{A},\mathcal{F})$ to be boundedly supermultiplicative. That is, there exist constants $C>0$ and $\alpha\ge0$, such that for all $n$, the number of words of length $n$ in $\mathcal{L}(\mathcal{A},\mathcal{F})$ is between $\alpha^n$ and $C\alpha^n$. In some settings, our condition provides a way to compute $C$, which implies that $\alpha$, the growth rate of the language, is also computable whenever our condition holds. We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to $\mathcal{F}$-free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words.

翻译：我们研究了 $\mathcal{L}(\mathcal{A},\mathcal{F})$ 增长率的一些性质，即字母表 $\mathcal{A}$ 上避免禁止因子集 $\mathcal{F}$ 的词语所构成的语言。我们首先给出了 $\mathcal{F}$ 和 $\mathcal{A}$ 的一个充分条件，使得 $\mathcal{L}(\mathcal{A},\mathcal{F})$ 的增长具有界超乘性。也就是说，存在常数 $C>0$ 和 $\alpha\ge0$，使得对于所有 $n$，$\mathcal{L}(\mathcal{A},\mathcal{F})$ 中长度为 $n$ 的词语数量介于 $\alpha^n$ 和 $C\alpha^n$ 之间。在某些设定下，我们的条件提供了一种计算 $C$ 的方法，这意味着只要我们的条件成立，语言增长率 $\alpha$ 也是可计算的。我们还将我们的技术应用于无幂词的具体设定，其中论证可以稍加精炼以提供更好的界。最后，我们将类似的思想应用于 $\mathcal{F}$-自由循环词，特别是我们在关于无平方循环词数量的 Shur 猜想方面取得了进展。