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

• 成果类型: Article
• 署名作者: Poels, Anthony
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2025.201.1.6
• 发表日期: 2025
• 页码: 307-330
• 关键词: parametric geometry

摘要

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 supremum omega & lowast;n(xi) of all omega for which infinitely many such alpha have |xi- alpha| <= H(alpha)-omega-1 is at least (n + 1)/2. He also asked if we could even have omega & lowast;n(xi) >= nas 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 omega & lowast;n(xi) >= anfor each n >= 4, with a = 1/root 3 similar or equal to 0.577. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that omega & lowast;n(xi) >= anfor each n >= 2, with a = 1/(2- log 2) similar or equal to 0.765.