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

Let ζ k be a k -th primitive root of unity, m ≥ø( k ) + 1 an integer and Φ k ( X ) ∈ ℤ[ X ] the k -th cyclotomic polynomial. In this paper we show that the pair (- m +ζ k , N ) is a canonical number system, with N = {0,1,...,|Φ k ( m )|-1}. Moreover we also discuss whether the two bases - m + ζ k and - n + ζ k are multiplicatively independent for positive integers m , n and k fixed.