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

Pour $\left\{{v}_{i}\right\}$ et $\left\{{w}_{j}\right\}$ deux familles de bases de ${\mathbf{C}}^{n}$, et $\theta$ un nombre fixe, nous considérons ${V}^{n}$ et ${W}^{n}$ deux sous-espaces engendrés par $\left[\theta ·n\right]$ vecteurs de $\left\{{v}_{i}\right\}$ et $\left\{{w}_{j}\right\}$ respectivement. Nous étudions l’angle entre ${V}^{n}$ et ${W}^{n}$ quand $n$ tend vers l’infini. Nous démontrons que, quand $\left\{{v}_{i}\right\}$ et $\left\{{w}_{j}\right\}$ 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.

If $\left\{{v}_{i}\right\}$ and $\left\{{w}_{j}\right\}$ are two families of unitary bases for ${\mathbf{C}}^{n}$, and $\theta$ is a fixed number, let ${V}^{n}$ and ${W}^{n}$ be subspaces of ${\mathbf{C}}^{n}$ spanned by $\left[\theta ·n\right]$ vectors in $\left\{{v}_{i}\right\}$ and $\left\{{w}_{j}\right\}$ respectively. We study the angle between ${V}^{n}$ and ${W}^{n}$ as $n$ goes to infinity. We show that when $\left\{{v}_{i}\right\}$ and $\left\{{w}_{j}\right\}$ 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.

@article{AIF_1987__37_1_175_0,
author = {Brooks, Robert},
title = {On the angles between certain arithmetically defined subspaces of ${\bf C}^n$},
journal = {Annales de l'Institut Fourier},
pages = {175--185},
publisher = {Institut Fourier},
volume = {37},
number = {1},
year = {1987},
doi = {10.5802/aif.1081},
zbl = {0611.15003},
mrnumber = {89h:11022},
language = {en},
url = {archive.numdam.org/item/AIF_1987__37_1_175_0/}
}
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. http://archive.numdam.org/item/AIF_1987__37_1_175_0/

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