Simultaneous approximation of algebraic irrationalities

This paper proves three theorems concerning the simultaneous approximation of numbers from a totally real algebraic number field. It is shown that for two given numbers $\theta_1$ and $\theta_2$ from a totally real algebraic number field, the constant $\gamma_{12}$ can be explicitly calculated, this being the upper limit of the numbers $C_{12}$ such that the inequality $\max(\|q\theta_1\|,\|q\theta_2\|)\leqslant(qC_{12})^{-\frac12}$ holds for infinitely many natural numbers $q$; likewise for the constant $a_{12}$ such that the inequality $\|q\theta_1\|\cdot\|q\theta_2\|<a_{12}(q^{\log}q)$ holds for infinitely many natural numbers $q$. It is shown that there exist $n-1$ numbers $\theta_1,\dots,\theta_{n-1}$ in an algebraic number field of degree n and discriminant d such that the inequality $\max(\|q\theta_1\|,\|q\theta_2\|)<(\gamma_q)^{-\frac{1}{n-1}}$ holds only for finitely many natural numbers $q$ if $\gamma>2^{-[\frac{n-1}{2}]}\sqrt{d}$ is fixed.