Anti-Power Prefixes of the Thue-Morse Word

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 $\displaystyle{\limsup_{k\to\infty}}(\Gamma(k)/k)=3$.