Antisquares and Critical Exponents

TL;DR

This work investigates antisquares in binary words to determine repetition thresholds for constrained word classes and to classify minimal antisquares. It introduces explicit morphic constructions, including $\mathbf{w} = g(\varphi^\omega(0))$, achieving a good-word critical exponent $\mathrm{ce}(\mathbf{w}) = 2+\alpha$, and proves this bound is optimal. The study then nicely partitions the binary-words landscape into the classes $AO_\ell$ and $AN_n$, providing exact threshold results via tailored morphisms $\xi_\ell$ and $\zeta_n$, and verifying with finite-state tools; it also yields a complete catalog of minimal antisquares and a tight growth-rate analysis for good words, identifying a sharp polynomial/exponential transition at $\tfrac{15}{4}$. Overall, the paper advances repetition-threshold theory in antisquare-avoiding words and connects combinatorial constructions with automatic methods and explicit enumerations. Further work proposes generalizations to morphic/antimorphic patterns and complements, broadening the scope of antisquare research.

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.

Figures (4)

• Figure 1: Fibonacci automaton for $\varphi^\omega(0)$.
• Figure 2: Fibonacci automaton for ${\bf w} = g(\varphi^\omega(0))$.
• Figure 3: Automaton for periods associated with $3^+$-powers in ${\bf w}$.
• Figure 4: Automaton for pairs $(n,p)$ associated with highest powers in ${\bf w}$.

Theorems & Definitions (47)

• Proposition 1
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof