Dirichlet improvability for $S$-numbers

TL;DR

This paper extends Dirichlet's theorem in metric Diophantine approximation to the $S$-adic setting over number fields, using dynamical systems techniques and nondivergence estimates.

Contribution

It provides a number field version of Dirichlet improvability results, generalizing previous work to arbitrary number fields and sets of places, including non-archimedean ones.

Findings

Established $S$-adic Dirichlet improvability results.

Generalized singularity of vectors to arbitrary number fields.

Extended dynamical methods to the $S$-adic setting.

Abstract

We study the problem of improving Dirichlet's theorem of metric Diophantine approximation in the $S$-adic setting. Our approach is based on translation of the problem related to Dirichlet improvability into a dynamical one, and the main technique of our proof is the $S$-adic version of quantitative nondivergence estimate due to D. Y. Kleinbock and G. Tomanov. The main result of this paper can be regarded as the number field version of earlier works of D. Y. Kleinbock and B. Weiss, and of the second named author and Anish Ghosh. Also this in turn generalises a result of Shreyasi Datta and M. M. Radhika on singularity of vectors to any number field $K$ and $S$ containing all archimedian places.