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- W2963456422 abstract "Recently, Fici, Restivo, Silva, and Zamboni defined a $k$-anti-power to be a word of the form $w_1w_2cdots w_k$, 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 $mathcal F(k)=AP({bf t},k)cap(2mathbb Z^+-1)$. For $kgeq 3$, $gamma(k)=min(mathcal F(k))$ and $Gamma(k)=max((2mathbb Z^+-1)setminusmathcal F(k))$ 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 $1/2leqdisplaystyle{liminf_{ktoinfty}}(gamma(k)/k)leq 9/10$ and $1leqdisplaystyle{limsup_{ktoinfty}}(gamma(k)/k)leq 3/2$. In addition, we prove that $displaystyle{liminf_{ktoinfty}}(Gamma(k)/k)=3/2$ and $displaystyle{limsup_{ktoinfty}}(Gamma(k)/k)=3$." @default.
- W2963456422 created "2019-07-30" @default.
- W2963456422 creator A5050123780 @default.
- W2963456422 date "2017-02-17" @default.
- W2963456422 modified "2023-09-23" @default.
- W2963456422 title "Anti-Power Prefixes of the Thue-Morse Word" @default.
- W2963456422 doi "https://doi.org/10.37236/6321" @default.
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