A Separation of $\gamma$ and $b$ via Thue--Morse Words

We prove that for $n\ge 2$, the size $b(t_n)$ of the smallest bidirectional scheme for the $n$th Thue–Morse word $t_n$ is $n+2$. Since Kutsukake et al. [SPIRE 2020] show that the size $\gamma (t_n)$ of the smallest string attractor for $t_n$ is $4$ for $n\ge 4$, this shows for the first time that there is a separation between the size of the smallest string attractor $\gamma$ and the size of the smallest bidirectional scheme $b$, i.e., there exist string families such that $\gamma =o(b)$.