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arXiv · 2005.09524

On repetitiveness measures of Thue-Morse words

Kanaru Kutsukake·Takuya Matsumoto·Yuto Nakashima·Shunsuke Inenaga·Hideo Bannai·Masayuki Takeda
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Abstract

We show that the size $\gamma(t_n)$ of the smallest string attractor of the $n$th Thue-Morse word $t_n$ is 4 for any $n\geq 4$, disproving the conjecture by Mantaci et al. [ICTCS 2019] that it is $n$. We also show that $\delta(t_n) = \frac{10}{3+2^{4-n}}$ for $n \geq 3$, where $\delta(w)$ is the maximum over all $k = 1,\ldots,|w|$, the number of distinct substrings of length $k$ in $w$ divided by $k$, which is a measure of repetitiveness recently studied by Kociumaka et al. [LATIN 2020]. Furthermore, we show that the number $z(t_n)$ of factors in the self-referencing Lempel-Ziv factorization of $t_n$ is exactly $2n$.

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Kanaru Kutsukake, Takuya Matsumoto, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda. 2020-05-19. On repetitiveness measures of Thue-Morse words. https://arxiv.org/abs/2005.09524

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