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.

Let p be a prime number. For a positive integer n and a real number ξ, let λn(ξ) denote the supremum of the real numbers λ for which there are infinitely many integer tuples (x 0 , x 1 , . . . , xn) such that |x 0 ξ−x 1 |p, . . . , |x 0 ξ n − xn|p are all less than X −λ−1 , where X is the maximum of |x 0 |, |x 1 |, . . . , |xn|. We establish new results on the Hausdorff dimension of the set of real numbers ξ for which λn(ξ) is equal to (or greater than or equal to) a given value.2010 Mathematics Subject Classification. 11J13.