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

TL;DR

This paper establishes an effective lower bound on the combined digit counts in two different base representations of algebraic integers in imaginary quadratic fields, extending Stewart's earlier work.

Contribution

It provides a new lower bound for the sum of non-zero digits in two distinct canonical number system expansions in imaginary quadratic fields, generalizing previous integer representation results.

Findings

Lower bound increases with the size of the algebraic integer

Effective bounds are derived for digit sums in two bases

Extension of Stewart's integer representation results to algebraic integers

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.