On multiplicatively independent bases in cyclotomic number fields

Recently the authors showed that the algebraic integers of the form $-m+{\zeta }_k$ are bases of a canonical number system of $\mathbb{Z}[{\zeta }_k]$ provided $m\ge \varphi (k)+1$, where ${\zeta }_k$ denotes a $k$-th primitive root of unity and $\varphi$ is Euler’s totient function. In this paper we are interested in the questions whether two bases $-m+{\zeta }_k$ and $-n+{\zeta }_k$ are multiplicatively independent. We show the multiplicative independence in case that $0<\mathrm{\mid }m-n\mathrm{\mid }<{10}^6$ and $\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }n\mathrm{\mid }>1$.