On badly approximable subspaces

TL;DR

This paper investigates the properties of badly approximable subspaces within a larger linear subspace, focusing on the behavior of almost all smaller subspaces in terms of Diophantine approximation.

Contribution

It provides new insights into the structure and approximation properties of subspaces contained within a given linear subspace in Euclidean space.

Findings

Almost all c-dimensional subspaces in the given subspace are badly approximable.

The study extends understanding of Diophantine approximation in higher-dimensional subspaces.

Results have implications for the geometry of numbers and Diophantine approximation theory.

Abstract

Given an $a$-dimensional linear subspace $\mathfrak{A}$ in $\mathbb{R}^d$ which contains a badly approximable $b$-dimensional subspace $\mathfrak{B} \subset \mathfrak{A}$. We study the badly approximability almost all $c$-dimensional linear subspaces $\mathfrak{C} \subset \mathfrak{A}$.