Simultaneous approximation of algebraic irrationalities

Abstract

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 θ1 and θ2 from a totally real algebraic number field, the constant γ12 can be explicitly calculated, this being the upper limit of the numbers c12 such that the inequality max (∥qθ1∥, ∥qθ2∥)⩽(qc12)−1/2 holds for infinitely many natural numbers q; likewise for the constant a12 such that the inequality ∥qθ1∥·∥qθ2∥ 2^{ - \left[ {\tfrac{{n - 1}}{2}} \right]} \sqrt d \). is fixed.