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

Abstract

Let $(\alpha ,{\mathcal{N}}_{\alpha })$ and $(\beta ,{\mathcal{N}}_{\beta })$ be two canonical number systems for an imaginary quadratic number field $K$ such that $\alpha$ and $\beta$ are multiplicatively independent. We provide an effective lower bound for the sum of the number of non-zero digits in the $\alpha$-adic and $\beta$-adic expansions of an algebraic integer $\gamma \in {\mathcal{O}}_K$ which is an increasing function of $|\gamma |$. This is an analogue of an earlier result due to Stewart on integer representations.

10 pages