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

3:00 pm, Friday, 23rd August

Abstract

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 exponent ω_n^*(ξ) which measures this quality of approximation 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 my talk, I will first present the background and ideas behind the proof of Wirsing's lower bound. Secondly, using a new approach that is partly inspired by parametric geometry of numbers, I will explain how we can obtain ω_n*(ξ) ≥ an for each n ≥ 2, with a = 1/(2 − log 2) ≃ 0.765.