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 $w^{(1)}, w^{(2)}, \ldots, w^{(k)}$ are distinct words of the
same length. For an infinite word $w$ and a positive integer $k$, define
$AP_j(w,k)$ 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 $\mathcal{F}_j(k) = (2 \mathbb{Z}^+ - 1) \cap AP_j(\mathbf{t},k)$,
where $\mathbf{t}$ denotes the Thue-Morse word. For all $k \in \mathbb{Z}^+$,
$\gamma_j(k) = \min (AP_j(\mathbf{t},k))$ 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 $\displaystyle 1/10 \leq \liminf_{k
\rightarrow \infty} (\gamma_j(k)/k) \leq 9/10$ and $\displaystyle 1/5 \leq
\limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2$. Additionally, we show
that $\displaystyle \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3/2$ and
$\displaystyle \limsup_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3$.
Comment: 21 pages
The Overlap Gap Between Left-Infinite and Right-Infinite Words
