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- W4300461268 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 $k$." @default.
- W4300461268 created "2022-10-03" @default.
- W4300461268 creator A5032573706 @default.
- W4300461268 date "2017-05-17" @default.
- W4300461268 modified "2023-10-03" @default.
- W4300461268 title "Functions on Antipower Prefix Lengths of the Thue-Morse Word" @default.
- W4300461268 doi "https://doi.org/10.48550/arxiv.1705.06310" @default.
- W4300461268 hasPublicationYear "2017" @default.
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