A simultaneous approximation problem for exponentials and logarithms

Abstract

Let $α_1,α_2$ be non-zero algebraic numbers such that $\frac{\log α_2}{\logα_1}\notin\mathbb{Q}$ and let $β$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $\left(\frac{\logα_2}{\log α_1},α_1^β, α_2^β\right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\logα_2}{\log α_1}$, $α^β_1$ and $α^β_2$ which are algebraically independent.