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Dirichlet improvability for $S$-numbers

Sourav Das, Arijit Ganguly

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.

Dirichlet improvability for $S$-numbers

Abstract

We study the problem of improving Dirichlet's theorem of metric Diophantine approximation in the -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 -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 and containing all archimedian places.
Paper Structure (7 sections, 13 theorems, 69 equations)

This paper contains 7 sections, 13 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Dirichlet's theorem over number fields
  3. Dirichlet improvability and the main theorem
  4. The correspondence
  5. Quantitative nondivergence and the proof of Proposition \ref{['main prop']}
  6. Quantitative nondivergence
  7. The proof of Proposition \ref{['main prop']}

Key Result

Theorem 1.1

For each $v \in S,$ let $\mathbf{y}^{(v)}=(y_1^{(v)},\dots,y_n^{(v)}) \in K_v^n.$ Also let $\varepsilon_v,$$\delta_v$$\in K_v \setminus \{0\}$ be given for each $v \in S$ with $|\varepsilon_v|_v <1$ and $|\delta_v|_v \geq 1$ for each $v \in S$ and Then there exist $\vec{\mathbf{q}}=(q_1,\dots,q_n) \in \mathcal{O}_S^n \setminus \{0\}$ and $p \in \mathcal{O}_S$ satisfying for all $v \in S.$

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 16 more