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

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