On approximation of real numbers by algebraic numbers of bounded degree

Dirichlet proved that for any real irrational number ξ there exist infinitely many rational numbers p/q such that |ξ−p/q|<q⁻². The correct generalization to the case of approximation by algebraic numbers of degree ?n, n>2, is still unknown. Here we prove a result which improves all previous estimates concerning this problem for n>2.