Rational approximation of affine coordinate subspaces of Euclidean space

TL;DR

This paper investigates the Diophantine approximation properties of affine coordinate subspaces in Euclidean space, establishing their Khintchine type for divergence under certain conditions and supporting a broader conjecture.

Contribution

It proves affine subspaces of dimension at least two are of Khintchine type for divergence and provides conditions for one-dimensional cases based on dual Diophantine type.

Findings

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

One-dimensional affine subspaces' approximation properties depend on the dual Diophantine type.

Evidence supporting the conjecture that all affine subspaces are of Khintchine type for divergence.

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 basepoint of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence.