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How can points be distributed uniformly over a sphere (following Lubotzky, Phillips and Sarnak). (Comment distribuer des points uniformément sur une sphère? (D’après Lubotzky, Phillips et Sarnak).) (French) Zbl 0900.11034
Séminaire de théorie spectrale et géométrie. Année 1986-1987. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Mathématiques, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 5, 9-18 (1987).
The author describes results of H. Kesten [Trans. Am. Math. Soc. 92, 336-354 (1959; Zbl 0092.33503)] as well as of A. Lubotzky, R. Phillips and P. Sarnak [Commun. Pure Appl. Math. 39, Suppl., S149–S186 (1986; Zbl 0619.10052); ibid. 40, 401-420 (1987; Zbl 0648.10034); Combinatorica 8, 261-277 (1988; Zbl 0661.05035); and Explicit expanders on the Ramanujan conjectures, Preprint Stanford Univ. (1986)]. The main focus is on a theorem of Lubotzky, et al.: For all \(S\subset SO(3)\) such that \(\#S= 2l\), \(\delta_S\geq 2\sqrt{2l-1}\), where \(\delta_S=\| T_{S_{| L_0^2(X)}}\|\) for \(T_S\) a particular operator defined on \(L^2(X)\), \(X\) being a compact, homogeneous Riemannian manifold.
For the entire collection see [Zbl 0825.00040].
Reviewer: Reviewer (Berlin)
MSC:
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11P21 Lattice points in specified regions
43A99 Abstract harmonic analysis
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
60G50 Sums of independent random variables; random walks
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