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 supremum $\omega ^*_n(\xi )$ 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 $a=1/(2-\log 2)\simeq 0.765$.