On approximation to a real number by algebraic numbers of bounded degree ; Sur l'approximation d'un nombre réel par des nombres algébriques de degré borné

Accepté pour publication dans Annals of Mathematics. ; International audience ; 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 ω_n*(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 ω_n* (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 ω_n* (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 ω_n* (ξ) ≥ an for each n ≥ 2, with a = 1/(2 -log 2) ≃ 0.765.