Two estimates concerning classical Diophantine approximation constants

Abstract

In this paper we aim to prove two inequalities involving the classical approximation constants $w_{n}^{\prime}(\zeta),\hat{w}_{n}^{\prime}(\zeta)$ that stem from the simultaneous approximation problem $|\zeta^{j}x-y_{j}|$, $1\leq j\leq n$, on the one side and the constants $w_{n}^{\ast}(\zeta),\hat{w}_{n}^{\ast}(\zeta)$ connected to approximation with algebraic numbers of degree $\leq n$ on the other side. We concretely prove $w_{n}^{\ast}(\zeta)\hat{w}_{n}^{\prime}(\zeta)\geq 1$ and $\hat{w}_{n}^{\ast}(\zeta)w_{n}^{\prime}(\zeta)\geq 1$. The first result is due to W. Schmidt, however our method of proving it allows to derive the other inequality as a dual result. Finally we will discuss estimates of $w_{n}^{\ast}(\zeta), \hat{w}_{n}^{\ast}(\zeta)$ uniformly in $\zeta$ depending only on $n$ as an application.