Approximation par des nombres algébriques

Abstract

For a real algebraic number θ of degree D , it follows from results of W. M. Schmidt and E. Wirsing that for every ε >0 and every positive integer d < D there exist infinitely many algebraic numbers α of degree d such that | θ − α |<H( α ) − d −1+ ε . Here, H denotes the naı̈ve height. In the present work, we provide very precise additional information about the height of such α 's. We also give a sharp approximation property valid for almost all real numbers (in the sense of Lebesgue measure) and show with an example that this cannot be satisfied by all real transcendental numbers. Further, as an application of our main theorem, we extend a previous result of E. Bombieri and J. Mueller in showing that, for any given real algebraic number θ , there exist infinitely many real number fields K for which precise information about effective approximation of θ relative to K can be given.