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Hecke operators and distributing points on the sphere. I. (English) Zbl 0619.10052
The authors present a very interesting and easily applicable method of constructing sequences on the sphere which are very well uniformly distributed. The method is based on choosing the points to be the orbit under the action of a countable subgroup $$\Gamma$$ of $$\text{SO}(3)$$ on $$S^ 2$$. Of particular importance are Hecke operators and Ramanujan’s conjecture (proved by Deligne).
The methods also give a (similar) proof of Drinfel’d’s theorem (Ruzewisz-Banach problem), i.e. there exist no nontrivial $$\text{SO}(3)$$-invariant mean on $$L^{\infty}(S^ 2)$$ [c.f. V. G. Drinfel’d, Funct. Anal. Appl. 18, 245–246 (1984); translation from Funkts. Anal. Prilozh. 18, No. 3, 77 (1984; Zbl 0576.28019)].
Using a classical result of H. Kesten [Trans. Am. Math. Soc. 92, 336–354 (1959; Zbl 0092.33503)], they obtain a lower bound for the operator discrepancy ($$\Gamma$$ acting on $$L^ 2(S^ 2)):\delta_{2N}>2N-1/N$$ for sequences $$\gamma_ 1,\gamma_ 2,\dots,\gamma_{2N}$$, $$\gamma_{N+1}=1/\gamma_ i$$. For any $$2N=p+1$$, $$p$$ a prime, there exist optimally distributed sequences (i.e. $$\delta_{2N}=\sqrt{2N}-1/N)$$ (For amenable $$\Gamma$$ : $$\delta_{2N}=1$$ for all $$N$$ !).
The second part of the paper contains results on the spherical cap discrepancy and on the mean-square discrepancy $$T$$. In particular, the authors study the case of $$\Gamma$$ generated by 3 rotations $$A, B, C$$ (corresponding to the quaternions $$1+2i$$, $$1+2j$$, $$1+2k$$ via $$\text{SU}(2)$$) and obtain $c_ 1/\sqrt{N}<T(\gamma_ 1x, \gamma_ 2x,\dots, \gamma_{2N}x)<c_ 2 \log N/\sqrt{N}$ for any fixed $$x\in S^ 2$$ (for arbitrary sequences in $$S^ 2$$ we have $$T(x_ 1,x_ 2,\dots,x_ n)\gg \varepsilon n^{-3/4-\varepsilon}$$ [W. M. Schmidt, Invent. Math. 7, 55–82 (1969; Zbl 0172.06402)]).
The final sections of the paper contain very instructive numerical computations and diagrams (in the special case of the rotations $$A, B, C$$).
Reviewer: H. Rindler

MSC:
 11K38 Irregularities of distribution, discrepancy 11F11 Holomorphic modular forms of integral weight 43A55 Summability methods on groups, semigroups, etc. 11K06 General theory of distribution modulo $$1$$ 47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces 60G50 Sums of independent random variables; random walks
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References:
  Cartier, Proc. Symp. In Pure Math., A.M.S. 26 pp 419– (1973)  Deligne, I, Publ. Math. IHES 43 pp 273– (1974) · Zbl 0287.14001  Drinfeld, Funct. Anal. and Applic. 18 pp 77– (1984)  and , Schottky Groups and Mumford Curves, S.L.N. 817, Springer 1980. · Zbl 0442.14009  Kesten, Trans. A.M.S. 92 pp 336– (1959)  Ergodic Properties of Algebraic Number Fields, Springer-Verlag, New York, 1968.  Rosenblatt, T.A.M.S. 265 pp 623– (1981)  Statistical properties of eigenvalues of the Hecke operators, to appear in the Proceedings of Number Theory Conf. Stillwater, 1984.  Schmidt, Inv. Math. 7 pp 55– (1969)  Fourier coefficients of Maass wave forms, Modular Forms, editor, John Wiley, New York, 1985, pp. 263–269.  Encrypting by random rotations, Cryptography, Lecture notes in Computer Science, 149, T. Beth editor, Berlin, 1983.  Szegö, A.M.S., Coll. Publ. (1939)
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