2024/05/14 by Poëls, Anthony
#11J13(Primary) #11J82 (Secondary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2405.08341
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 infimum 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.