Antisquares and Critical Exponents

Abstract

The complement $\bar{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An antisquare is a nonempty word of the form $x\, \bar{x}$. 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 large number. We say a binary word is 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 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