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 $|{\zeta }^{j}x-y_j|$, $1\le j\le n$, on the one side and the constants $w_n^{\ast }(\zeta ),{\hat{w}}_n^{\ast }(\zeta )$ connected to approximation with algebraic numbers of degree $\le n$ on the other side. We concretely prove $w_n^{\ast }(\zeta ){\hat{w}}_n^{\prime }(\zeta )\ge 1$ and ${\hat{w}}_n^{\ast }(\zeta )w_n^{\prime }(\zeta )\ge 1$. 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.

arXiv comment11 pages