A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article, we give a structure theorem for infinite binary rich words that avoid $14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.

中文翻译：

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二进制丰富词的重复阈值

如果长度为 $n$ 的单词包含 $n$ 非空回文因子，则它是丰富的。如果一个无限词的所有有限因素都是丰富的，那么它就是丰富的。Baranwal 和 Shallit 产生了一个具有临界指数 $2+\sqrt{2}/2$ ($\approx 2.707$) 的无限富二元词，并推测这是无限富二元词的最小临界指数（即重复二进制丰富词的阈值是 $2+\sqrt{2}/2$)。在本文中，我们给出了避免$14/5$-powers（即指数至少为2.8 的重复）的无限二元富词的结构定理。因此，我们推断出二进制丰富词的重复阈值为 $2+\sqrt{2}/2$，正如 Baranwal 和 Shallit 所推测的那样。这解决了二进制字母表 Vesti 的一个开放问题；对于更大的字母，这个问题仍然存在。