Matrix periods and competition periods of Boolean Toeplitz matrices II

This paper is a follow-up to the paper of Cheon et al. (2023) [2]. Given subsets S and T of $\{1,\dots ,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n〈S;T〉$ is defined to have 1 as the $(i,j)$-entry if and only if $j-i\in S$ or $i-j\in T$. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices $A=T_n〈S;T〉$ satisfying the condition (⋆) $\max S+\min T\le n$ and $\min S+\max T\le n$ are $d^{+}/d$ and 1, respectively, where $d^{+}=\gcd (s+t|s\in S,t\in T)$ and $d=\gcd (d,\min S)$. In this paper, we claim that even if (⋆) is relaxed to the existence of elements $s\in S$ and $t\in T$ satisfying $s+t\le n$ and $\gcd (s,t)=1$, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy (⋆) but the relaxed condition. For example, for any positive integers $k,n$ with $2k+1\le n$, it is easy to see that $T_n〈k,n-k;k+1,n-k-1〉$ does not satisfy (⋆) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence ${\left\{A^m{\left(A^T\right)}^m\right\}}_{m=1}^{\infty }$ is $T_n〈d^{+},2d^{+},\dots ,\lfloor n/d^{+}\rfloor d^{+}〉$.