An improved bound in Wirsing's problem

TL;DR

This paper improves the lower bound for the approximation exponent related to Wirsing's problem, surpassing previous bounds and establishing new relations among classical exponents of approximation.

Contribution

It provides a new lower bound exceeding $n/\sqrt{3}$ for the approximation exponent and introduces relations between classical exponents.

Findings

Lower bound exceeds $n/\sqrt{3}$ for the approximation exponent.

Provides new relations between classical exponents of approximation.

Offers a qualitative improvement over previous bounds.

Abstract

We improve the lower bound for the classical exponent of approximation $w_{n}^{\ast}(\xi)$ connected to Wirsing's famous problem of approximation to real numbers by algebraic numbers of degree at most $n$. Our bound exceeds $n/\sqrt{3}\approx 0.5773n$ and thus provides a reasonable qualitative improvement to previous bounds of order $n/2+O(1)$. We further establish new relations between several classical exponents of approximation.