Antisquares and Critical Exponents

TL;DR

This paper investigates infinite binary words avoiding large antisquares, characterizes minimal antisquares, and analyzes the growth and repetition thresholds of 'good' words that contain only the smallest antisquares.

Contribution

It introduces the concept of good words, characterizes minimal antisquares, and determines growth rates and thresholds related to antisquare-free binary words.

Findings

Repetition threshold for binary words with exactly two antisquares is (5+√5)/2.

Characterization of minimal antisquares and their properties.

Determination of growth rate and threshold between polynomial and exponential growth.

Abstract

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.