Simultaneous Diophantine approximation on affine subspaces and Dirichlet improvability

TL;DR

This paper investigates Diophantine approximation properties of affine subspaces, establishing their Khintchine type for divergence, and explores related weighted approximation and Dirichlet improvability results, advancing understanding of these approximation phenomena.

Contribution

It proves affine subspaces of dimension at least two are of Khintchine type for divergence and extends results to weighted approximation and Dirichlet improvability.

Findings

Affine subspaces of dimension ≥2 are of Khintchine type for divergence.

Weighted badly approximable vectors are weighted Dirichlet improvable.

Relations between non-singularity and twisted inhomogeneous approximation are established.

Abstract

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the base point of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence. We also prove a partial analogue regarding the Hausdorff measure theory. Furthermore, we obtain various results relating weighted Diophantine approximation and Dirichlet improvability. In particular, we show that weighted badly approximable vectors are weighted Dirichlet improvable, thus generalising a result by Davenport and Schmidt. We also provide a relation between non-singularity and twisted inhomogeneous approximation. This extends a result of Shapira to the weighted case.