Two estimates concerning classical Diophantine approximation constants

TL;DR

This paper establishes two inequalities relating classical Diophantine approximation constants, providing new proofs and dual results, and discusses uniform estimates depending only on the degree n.

Contribution

The paper proves two key inequalities involving approximation constants, offering a novel proof for one and deriving a dual inequality, advancing understanding in Diophantine approximation theory.

Findings

Proved that $w_{n}^{ ext{*}}(zeta)\, ext{and}\, ilde{w}_{n}^{ ext{*}}(zeta)$ satisfy specific inequalities.

Derived dual inequalities connecting different approximation constants.

Discussed uniform estimates of these constants depending only on the degree n.

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.