On Some Properties of Irrational Subspaces

TL;DR

This paper explores properties of completely irrational subspaces, demonstrating their approximation characteristics, bounds on Diophantine exponents, and the winning nature of sets of such subspaces.

Contribution

It introduces new properties of completely irrational subspaces, including their approximation behavior and bounds on Diophantine exponents, advancing understanding in Diophantine approximation.

Findings

Existence of badly approximable completely irrational subspaces.

Sets of such subspaces are winning in various senses.

Bounds for Diophantine exponents of vectors in these subspaces.

Abstract

In this paper we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector $\xi$ from two-dimensional badly approximable completely irrational subspace of $\mathbb{R}^d$ one has $\hat{\omega}(\xi) \leq \frac{\sqrt{5} - 1}{2}$. Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.