Going-up theorems for simultaneous Diophantine approximation

Abstract

We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector $\underline{\xi}=(\xi,\xi^{2},\ldots,\xi^{N})$, in various dimensions $N$. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with $\mathbb{Q}$-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, applied in the framework of parametric geometry of numbers introduced by Schmidt and Summerer. We also frequently employ Mahler's Duality result on polar convex bodies.