On the angles between certain arithmetically defined subspaces of \({\mathbb C}^ n\) (Q1087608)

If \(\{v_ i\}\) and \(\{w_ j\}\) are two families of unitary bases for \({\mathbb{C}}^ n\), and \(\theta\) is a fixed number, let \(V^ n\) and \(W^ n\) be subspaces of \({\mathbb{C}}^ n\) spanned by [\(\theta\) \(\cdot n]\) vectors in \(\{v_ i\}\) and \(\{w_ j\}\), respectively. We study the angle between \(V^ n\) and \(W^ n\) as n goes to infinity. We show that when \(\{v_ i\}\) and \(\{w_ j\}\) arise in certain arithmetically defined families, the angles between \(V^ n\) and \(W^ n\) may either tend to 0 or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.