Matrix periods and competition periods of Boolean Toeplitz matrices II

This paper is a follow-up to the paper [Matrix periods and competition periods of Boolean Toeplitz matrices, {\it Linear Algebra Appl.} 672:228--250, (2023)]. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ 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 $A=T_n\langle S; T \rangle$ satisfying the condition ($\star$) $\max S+\min T \le n$ and $\min S+\max T \le n$ are $d^+/d$ and $1$, respectively, where $d^+= \gcd (s+t \mid s \in S, t \in T)$ and $d = \gcd(d, \min S)$. In this paper, we claim that even if ($\star$) 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 ($\star$) but the relaxed condition. For example, for any positive integers $k, n$ with $2k+1 \le n$, it is easy to see that $T_n\langle k, n-k;k+1, n-k-1 \rangle$ does not satisfies ($\star$) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is $T_n\langle d^+,2d^+, \ldots, \lfloor n/d^+\rfloor d^+\rangle$.