An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields

Let ${\zeta }_k$ be a $k$-th primitive root of unity, $m\ge \varphi (k)+1$ an integer and ${\mathrm{\Phi }}_k(X)\in \mathbb{Z}[X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+{\zeta }_k,\mathcal{N})$ is a canonical number system, with $\mathcal{N}=\{0,1,\dots ,\mathrm{\mid }{\mathrm{\Phi }}_k(m)\mathrm{\mid }\}$. Moreover we also discuss whether the two bases $-m+{\zeta }_k$ and $-n+{\zeta }_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.