Anti-Power Prefixes of the Thue-Morse Word

TL;DR

This paper investigates the growth rates of anti-power prefixes in the Thue-Morse word, proving linear bounds on the minimal and maximal odd-length prefixes that form anti-powers.

Contribution

It establishes that the minimal anti-power prefix length ratios grow linearly with k, confirming a conjecture and providing bounds for their asymptotic behavior.

Findings

ici et al.'s conjecture on linear growth is confirmed.

ici et al.'s conjecture on linear growth is confirmed.

The asymptotic ratios for (k) are bounded between 1/2 and 3/2.

Abstract

Recently, Fici, Restivo, Silva, and Zamboni defined a $k$-anti-power to be a word of the form $w_1w_2\cdots w_k$, where $w_1,w_2,\ldots,w_k$ are distinct words of the same length. They defined $AP(x,k)$ to be the set of all positive integers $m$ such that the prefix of length $km$ of the word $x$ is a $k$-anti-power. Let ${\bf t}$ denote the Thue-Morse word, and let $\mathcal F(k)=AP({\bf t},k)\cap(2\mathbb Z^+-1)$. For $k\geq 3$, $\gamma(k)=\min(\mathcal F(k))$ and $\Gamma(k)=\max((2\mathbb Z^+-1)\setminus\mathcal F(k))$ are well-defined odd positive integers. Fici et al. speculated that $\gamma(k)$ grows linearly in $k$. We prove that this is indeed the case by showing that $1/2\leq\displaystyle{\liminf_{k\to\infty}}(\gamma(k)/k)\leq 9/10$ and $1\leq\displaystyle{\limsup_{k\to\infty}}(\gamma(k)/k)\leq 3/2$. In addition, we prove that $\displaystyle{\liminf_{k\to\infty}}(\Gamma(k)/k)=3/2$ and $\displaystyle{\limsup_{k\to\infty}}(\Gamma(k)/k)=3$.