An improved bound in Wirsing’s problem

Abstract

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