  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 $a=1/(2-\log 2)\simeq
0.765$.
