On Some Properties of Irrational Subspaces
Vasiliy Neckrasov
TL;DR
The paper studies completely irrational subspaces and their badly approximable variants using Schmidt-type games. It proves the existence of completely irrational badly approximable subspaces and establishes that the associated sets $I(n,m)$ and $N^{irr}(n,m)$ are winning in Schmidt-like games, including hyperplane absolute winning, which implies full fractal dimension properties. It also derives Diophantine-exponent bounds for vectors lying in two-dimensional badly approximable completely irrational subspaces, notably $\hat{\omega}(\xi) \le \frac{\sqrt{5}-1}{2}$, and develops a generalized manifold-escaping lemma for algebraic manifolds. Together, these results elucidate the metric and structural richness of badly approximable and completely irrational subspaces and their interactions under dynamical game frameworks.
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 $ξ$ from two-dimensional badly approximable completely irrational subspace of $\mathbb{R}^d$ one has $\hatω(ξ) \leq \frac{\sqrt{5} - 1}{2}$. Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.