Anti-Power Prefixes of the Thue-Morse Word

Abstract

Recently, Fici, Restivo, Silva, and Zamboni defined a $k$-anti-power to be a word of the form $w_1{w}_2\cdots w_k$, where $w_1,w_2,\dots ,w_k$ 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 $\mathbf{t}$ denote the Thue-Morse word, and let $\mathcal{F}(k)=AP(\mathbf{t},k)\cap (2{\mathbb{Z}}^{+}-1)$. For $k\ge 3$, ${\hat{I}}^3(k)=\min (\mathcal{F}(k))$ and $\hat{I}“(k)=\max ((2{\mathbb{Z}}^{+}-1)\setminus \mathcal{F}(k))$ are well-defined odd positive integers. Fici et al. speculated that ${\hat{I}}^3(k)$ grows linearly in $k$. We prove that this is indeed the case by showing that $1/2\le \underset{k\to \infty }{\mathrm{liminf}}({\hat{I}}^3(k)/k)\le 9/10$ and $1\le \underset{k\to \infty }{\mathrm{limsup}}({\hat{I}}^3(k)/k)\le 3/2$. In addition, we prove that $\underset{k\to \infty }{\mathrm{liminf}}(\hat{I}“(k)/k)=3/2$ and $\underset{k\to \infty }{\mathrm{limsup}}(\hat{I}“(k)/k)=3$.

15 pages, 4 figures