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 integ er d < D there exist infinitely many algebraic numbers alpha of degree d su ch that \0-alpha\ < H(alpha)(-d-1+epsilon). Here, H denotes the naive heigh t. In the present work, we provide very precise additional information abou t the height of such alpha's. We also give a sharp approximation property v alid fbr almost all real numbers tin the sense of Lebesgue measure) and sho w with an example that this cannot he satisfied by all real transcendental numbers. Further, as an application of our main theorem, we extend a previo us 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 wh ich precise information about effective approximation of 0 relative to K ca n be given. (C) 2000 Academic Press.