On the approximation to algebraic numbers by algebraic numbers

Abstract

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number e, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+e, where H(α) denotes the naive height of α. We sharpen this result by replacing e by a function H 7→ e(H) that tends to zero when H tends to infinity. We make a similar improvement for the approximation to algebraic numbers by algebraic integers, as well as for an inhomogeneous approximation problem.