| Summary: | 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.
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