Anti-power $j$-fixes of the Thue-Morse word

Abstract

Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a $k$-anti-power, which is defined as 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. For an infinite word $w$ and a positive integer $k$, define $A{P}_j(w,k)$ to be the set of all integers $m$ such that $w_{j+1}{w}_{j+2}\cdots w_{j+km}$ is a $k$-anti-power, where $w_i$ denotes the $i$-th letter of $w$. Define also ${\mathcal{F}}_j(k)=(2{\mathbb{Z}}^{+}-1)\cap A{P}_j(\mathbf{t},k)$, where $\mathbf{t}$ denotes the Thue-Morse word. For all $k\in {\mathbb{Z}}^{+}$, ${\gamma }_j(k)=\min (A{P}_j(\mathbf{t},k))$ is a well-defined positive integer, and for $k\in {\mathbb{Z}}^{+}$ sufficiently large, ${\Gamma }_j(k)=\sup ((2{\mathbb{Z}}^{+}-1)\setminus {\mathcal{F}}_j(k))$ is a well-defined odd positive integer. In his 2018 paper, Defant shows that ${\gamma }_0(k)$ and ${\Gamma }_0(k)$ grow linearly in $k$. We generalize Defant's methods to prove that ${\gamma }_j(k)$ and ${\Gamma }_j(k)$ grow linearly in $k$ for any nonnegative integer $j$. In particular, we show that $1/10\le \underset{k\to \infty }{\mathrm{liminf}}({\gamma }_j(k)/k)\le 9/10$ and $1/5\le \underset{k\to \infty }{\mathrm{limsup}}({\gamma }_j(k)/k)\le 3/2$. Additionally, we show that $\underset{k\to \infty }{\mathrm{liminf}}({\Gamma }_j(k)/k)=3/2$ and $\underset{k\to \infty }{\mathrm{limsup}}({\Gamma }_j(k)/k)=3$.

21 pages