Effective simultaneous rational approximation to pairs of real quadratic numbers

Abstract

Let $\xi$, $\zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$ such that, for every integer $q$ with $q>c$, we have

$$max\left\{\parallel q\xi \parallel ,\parallel q\zeta \parallel \right\}>q^{-1+\tau },$$

where $\parallel \cdot \parallel$ denotes the distance to the nearest integer.