Functions on Antipower Prefix Lengths of the Thue-Morse Word

Abstract

We say that a word $w$ of length $kn$ is a $k$-\textit{antipower} if it can be written in the form $w_1 \cdots w_k$, where each $w_i$ is a distinct word of length $n$. We analyze prefixes of the Thue-Morse word $\textbf{t}$ and lengths of antipowers occurring in them. Define $\Gamma(k)$ to be the largest odd $n$ such that the prefix of $\textbf{t}$ of length $kn$ is not a $k$-antipower, and $\gamma(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $\gamma(k)$ and $\Gamma(k)-\gamma(k)$. Our bounds on $\gamma(k)$ affirmatively answer one conjecture of Defant and make substantial progress towards answering a second conjecture of Defant. It was previously known that $\Gamma(k)$ and $\gamma(k)$ grow linearly in $k$, but our bounds on $\Gamma(k)-\gamma(k)$ prove that $\Gamma(k)-\gamma(k)$ also grows linearly in $k$.