Approximation by algebraic numbers

For a real algebraic number 0 of degree D, it follows from results of W. M. Schmidt and E. Wirsing that For every epsilon > 0 and every positive integer d < D there exist infinitely many algebraic numbers alpha of degree d such that \0-alpha\ < H(alpha)(-d-1+epsilon). Here, H denotes the naive height. In the present work, we provide very precise additional information about the height of such alpha's. We also give a sharp approximation property valid fbr almost all real numbers tin the sense of Lebesgue measure) and show with an example that this cannot he 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 ant given oat algebraic number 0, there exist infinitely many real number fields K for which precise information about effective approximation of 0 relative to K can be given. (C) 2000 Academic Press.