论文标题
反quar和关键指数
Antisquares and Critical Exponents
论文作者
论文摘要
通过将$ x $ x $的$ x $的（x} $（x} $）更改为$ x $的（$ x $），$ x $ in $ x $ in $ x $的（x} $）的（bitwise）补充$ \ overline {x} $，反之亦然。 $ \ textit {antisquare} $是$ x \，\ overline {x} $的非空词。在本文中，我们研究了不包含任意大型反quar的无限二进制词。例如，我们表明，完全包含两个不同的抗Quares的无限二进制单词语言的重复阈值是$（5+ \ sqrt {5} {5}）/2 $。我们还研究了相关类别的重复阈值，在上一句话中，“两个”被较大数字所取代。 我们说，如果其唯一包含的反quares为$ 01 $和$ 10 $，则二进制单词是$ \ textit {good} $。我们表征了最小的反quar虫，也就是说，那些是反quar的词，但所有适当的因素都是好的。我们确定长度$ n $的好单词数量的增长率，并确定好词数的多项式和指数增长之间的重复阈值。
The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.
