On some properties of irrational subspaces (Q2146244)

The author recalls the concepts of rational subspaces, irrational and badly approximable subspaces. Also, briefly describes a classical game introduced by \textit{W. M. Schmidt}, [Diophantine approximation. Springer-Verlag (1980; Zbl 0421.10019)]. One of the main results of the paper is the following:\par Theorem 2.1: Suppose that \(\alpha\) and \(\beta\) satisfy \(2\alpha\leq 1+\alpha\beta\). Then the set \(I(n,m)\) of all \(n \times m\)-matrices \(\theta\) defining \(n\)-dimensional completely irrational subspace of the form \(\mathcal L=\{(x,y):x\in\mathbb R^n, y=\theta x\}\) in \(\mathbb R^{n+m}\) is \((\alpha, \beta)\)-winning. In particular \(I(n,m)\) is \(\alpha\)-winning for \(\alpha\le 1/2\).