Simultaneous approximation in number fields
Let \(k\) be a number field and let \(S\) be a non-empty finite set of places of \(k\). The theorem of independence of places shows that the diagonal embedding maps \(k\) to a dense subset of the product of its completions \(k_ v\) at the places \(v\) of \(S\). In particular, the multiplicative group \(k^ \times\) of \(k\) has a dense image in the product \(G\) of the multiplicative groups \(k_ v^ \times\) for all \(v\in S\). When \(S\) is the set \(S_ \infty\) of all archimedean places of \(k\), the group \(G\) is isomorphic to \((\mathbb{R} \otimes_ \mathbb{Q} k)^ \times\) and \textit{M. Waldschmidt} showed the existence of a finitely generated subgroup of \(k^ \times\) whose image is dense in \(G\) [Théorie des Nombres, Sémin. Delange-Pisot-Poitou, Paris 1980-81, Prog. Math. 22, 323-335 (1982; Zbl 0494.12007)]. The present paper shows more generally that if \(m\) denotes the smallest integer for which there exists a dense subgroup of \(G\) generated by \(m\) elements, then there exist \(m\) elements of \(k^ \times\) whose images in \(G\) generate a dense subgroup of \(G\). When \(S\subset S_ \infty\), the integer \(m(G)\) is \(| S|+1\) and this result answers a problem of \textit{J. J. Sansuc} [Théorie des Nombres, Sémin. Delange-Pisot-Poitou, Paris 1980-81, Prog. Math. 22, 253-271 (1982; Zbl 0506.14014)] about the smallest rank of a finitely generated dense subgroup of \((\mathbb{R}\otimes_ \mathbb{Q} k)^ \times\) inside \(k^ \times\). This rank is \(| S_ \infty|+1\). The proof relies on a transcendence result of \textit{M. Waldschmidt} [Invent. Math. 63, 97-127 (1981; Zbl 0454.10020)] and on a general fact about finitely generated subgroups of connected commutative real Lie groups.
scientific article; zbMATH DE number 220749