Singular Vectors on Manifolds over totally real Number Fields

Abstract

We extend the notion of singular vectors in the context of Diophantine approximation of real numbers with elements of a totally real number field $K$. For $m\geq1$, we establish a version of Dani's correspondence in number fields and prove that under a class of `friendly measures' in $K_S^m$, the set of singular vectors has measure zero. Here $S$ is the set of Archimedean valuations of $K$ and $K_S$ is the product of the completions of $\sigma(K)$, $\sigma\in S$. On the other hand, we show the existence of uncountably many non-trivial singular vectors on suitable submanifolds of $K^m_S$ under the action of a certain one parameter subgroup of $\mathrm{SL}_{m+1}(K_S)$.