A simultaneous approximation problem for exponentials and logarithms

Abstract

Let $\hat{I}{\pm }_1,\hat{I}{\pm }_2$ be non-zero algebraic numbers such that $\frac{\log \hat{I}{\pm }_2}{\log \hat{I}{\pm }_1}\notin \mathbb{Q}$ and let ${\hat{I}}^2$ 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 $\left(\frac{\log \hat{I}{\pm }_2}{\log \hat{I}{\pm }_1},\hat{I}{\pm }_1^2,\hat{I}{\pm }_2^2\right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\log \hat{I}{\pm }_2}{\log \hat{I}{\pm }_1}$, $\hat{I}{\pm }_1^2$ and $\hat{I}{\pm }_2^2$ which are algebraically independent.

20 pages. Comments are welcome