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∥< a₁₂(q^logq) holds for infinitely many natural numbers q. It is shown that there exist n −1 numbers θ₁, ..., θ_n−1 in an algebraic number field of degree n and discriminant d such that the inequality$\max \left( {\left\| {q\theta _1 } \right\|, ..., \left\| {q\theta _{n - 1} } \right\|} \right)< \left( {\gamma q} \right)^{ - \frac{1}{{n - 1}}}$ holds only for finitely many natural numbers q if$\gamma > 2^{ - \left[ {\tfrac{{n - 1}}{2}} \right]} \sqrt d$. is fixed.