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 $$ \max\{\|q \xi \|, \|q \zeta\| \} > q^{-1 + \tau}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.