On the representation of an imaginary quadratic integer in two different bases

Abstract

Let (α,Nα) and (β,Nβ) be two canonical number systems for an imaginary quadratic number field K such that α and β are multiplicatively independent. We provide an effective lower bound for the sum of the number of non-zero digits in the α-adic and β-adic expansions of an algebraic integer γ∈OK which is an increasing function of |γ|. This is an analogue of an earlier result due to Stewart on integer representations