An improved bound in Wirsing’s problem

Badziahin, Dmitry; Schleischitz, Johannes

doi:10.1090/tran/8245

An improved bound in Wirsing’s problem
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by Dmitry Badziahin and Johannes Schleischitz PDF

Trans. Amer. Math. Soc. 374 (2021), 1847-1861 Request permission

Abstract:

We improve the lower bound for the classical exponent of approximation $w_{n}^{\ast }$ connected to Wirsing’s famous problem on approximation to real numbers by algebraic numbers of degree at most $n$. Our bound exceeds $n/\sqrt {3}\approx 0.5773n$ and thus provides a considerable qualitative improvement to previous bounds of order $n/2+O(1)$. We further establish new relations between several classical exponents of approximation.

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