On approximation of real numbers by algebraic numbers of bounded degree

Abstract

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 .