Galois generators and the subspace theorem

Abstract

In an earlier paper [2], a preliminary attempt was made to investigate the divisibility properties of the set of normal integral generators in a tame, prime extension ofQ. The result suggested that the contribution from an arbitrary, fixed, finite set of primes is very small. In this paper we are going to show (1) that it is about as small as it could be and (2) that the results hold in general for tame, abelian extensions (with only a mild technical condition). The approach is twofold, depending upon (a) a good understanding of the geometry of the units in abelian group rings, and (b) the generalisation, due to Schlickewei, of the beautiful and powerful subspace theorem of W. Schmidt. In previous papers, attempts to use diophantine approximation for these problems have been frustrated by the unwieldy nature of the theorems involved. Here we go to the heart of the matter by making an appeal directly to the subspace theorem.