On Heights of Algebraic Subspaces and Diophantine Approximations

Abstract

The basic problem of diophantine approximations could be formulated follows. Assume a Cartesian coordinate system in euclidean E2 to be given. Let L be a line through the origin 0. Find a line L' through 0 and defined over the rationals Q (defined by a linear equation with rational coefficients) which is as possible to L. As a measure of their closeness, introduce /r(L, L') = sin Ap, where p is the angle between L and L', 0 0, there will be infinitely many such lines L' satisfying J(L, L') 0 and a line L through 0 which is not defined over the rationals, there are infinitely many rational lines L' through 0 satisfying !(L, L') < (5-1/2 + 8) H(L)-2 This statement would become false if 5-1/2 were replaced by a smaller constant. The problem of simultaneous approximation amounts to the problem of finding a line L' of euclidean En through the origin 0 and defined over the rationals which is close to a given line through 0. The problem of linear forms amounts to finding a rational hyperplane H of euclidean En through 0 and close to a given line through 0. This suggests the following problem. Let K be a fixed real algebraic number field of finite degree over the rationals Q. Let 0 < d < n, 0 < e < n, and A a d-dimensional subspace of euclidean En. Find a subspace B of En of dimension e and defined over K, such that B is to A. One can also