On approximation to a real number by algebraic numbers of bounded degree

TL;DR

This paper improves lower bounds on how well transcendental real numbers can be approximated by algebraic numbers of bounded degree, using a new approach inspired by parametric geometry of numbers.

Contribution

It establishes a new lower bound for approximation exponents for all degrees n ≥ 2, surpassing previous results by Badziahin and Schleischitz.

Findings

New lower bound: ω*_n(ξ) ≥ a·n with a ≈ 0.765

Improved bounds hold for all n ≥ 2

Approach inspired by parametric geometry of numbers

Abstract

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $\xi$ can be approximated by algebraic numbers $\alpha$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(\alpha)$ of $\alpha$. He showed that the infimum $\omega^*_n(\xi)$ of all $\omega$ for which infinitely many such $\alpha$ have $|\xi-\alpha| \le H(\alpha)^{-\omega-1}$ is at least $(n+1)/2$. He also asked if we could even have $\omega^*_n(\xi) \ge n$ as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form $n/2+\mathcal{O}(1)$ until Badziahin and Schleischitz showed in 2021 that $\omega^*_n(\xi) \ge an$ for each $n\ge 4$, with $a=1/\sqrt{3}\simeq 0.577$. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that $\omega^*_n(\xi) \ge an$ for each $n\ge 2$, with $a=1/(2-\log 2)\simeq 0.765$.