no. 1
On the angles between certain arithmetically defined subspaces of 𝐂 n
Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 175-185

If {v i } and {w j } are two families of unitary bases for C n , and θ is a fixed number, let V n and W n be subspaces of C n spanned by [θ·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.

Pour {v i } et {w j } deux familles de bases de C n , et θ un nombre fixe, nous considérons V n et W n deux sous-espaces engendrés par [θ·n] vecteurs de {v i } et {w j } respectivement. Nous étudions l’angle entre V n et W n quand n tend vers l’infini. Nous démontrons que, quand {v i } et {w j } sont présents dans certaines familles définies arithmétiquement, l’angle entre V n et W n peut soit tendre vers 0, soit être minoré par une constante strictement positive. Ce comportement dépend d’un problème de valeur propre associé.

Brooks, Robert. On the angles between certain arithmetically defined subspaces of ${\bf C}^n$. Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 175-185. doi: 10.5802/aif.1081
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     title = {On the angles between certain arithmetically defined subspaces of ${\bf C}^n$},
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