An improved bound in Wirsing's problem

2019/12/19 by Badziahin, Dzmitry, Schleischitz, Johannes
#11J13 #11J82 #11J83 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.09013

Abstract

We improve the lower bound for the classical exponent of approximation w_n^∗(ξ) connected to Wirsing's famous problem of approximation to real numbers by algebraic numbers of degree at most n. Our bound exceeds n/√(3)≈ 0.5773n and thus provides a reasonable qualitative improvement to previous bounds of order n/2+O(1). We further establish new relations between several classical exponents of approximation.

Related

• Simultaneous rational approximation to successive powers of a real number
• On approximation to a real number by algebraic numbers of bounded degree
• On uniform polynomial approximation
• Wirsing-type inequalities
• Exponents of Diophantine approximation