On multiplicatively independent bases in cyclotomic number fields

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

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