Singular Vectors on Manifolds over totally real Number Fields

TL;DR

This paper extends the concept of singular vectors to Diophantine approximation over totally real number fields, establishing measure-zero results for singular vectors and demonstrating their existence on certain submanifolds.

Contribution

It generalizes Dani's correspondence to number fields and proves measure-zero for singular vectors under friendly measures, while also constructing uncountably many singular vectors on submanifolds.

Findings

Singular vectors have measure zero under friendly measures in number fields.

Existence of uncountably many non-trivial singular vectors on submanifolds.

Extension of Dani's correspondence to 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 $\mathrm{SL}_{m+1}(K_S)$.