We prove an inequality for the number of periods in a word x in terms of the length of x and its initial critical exponent. Next, we characterize all periods of the length-n prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of n, and use this result to show that our inequality is tight for infinitely many words x. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where x is overlap-free or squarefree.

中文翻译：

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一个单词中句点数的不等式

我们根据 x 的长度及其初始临界指数证明了单词 x 中句点数的不等式。接下来，我们根据 n 的惰性 Ostrowski 表示来表征一个特征 Sturmian 词的长度为 n 前缀的所有周期，并使用这个结果来表明我们的不等式对于无限多个词 x 是紧密的。我们为无限词提出了两种相关的周期性度量。最后，我们还考虑 x 是无重叠或无平方的特殊情况。