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Hecke operators and distributing points on $$S^2$$. II. (English) Zbl 0648.10034
In the very interesting first part [ibid. 39, Special issue, S149–S186 (1986; Zbl 0619.10052)] the authors showed how to construct very evenly distributed sequences on the sphere $$S^ 2$$, considering Hecke operators $$T$$ on $$L^ 2(S^ 2)$$. Of essential importance is an upper bound on the absolute value of the eigenvalue having the second-largest modulus (denoted by $$\lambda_ 1(T)$$).
Let $$S_ p$$ be the set of all quaternions $$\alpha =a_ 0+a_ 1i+a_ 2j+a_ 3k$$, $$a_ i\in {\mathbb Z}$$ with norm $$N(\alpha)=p$$, $$a_ 0>0$$, $$\alpha\equiv 1\pmod 2$$ ($$\text{card}\,S_ p=p+1)$$. If $$T_ p: L^ 2(S^ 2)\to L^ 2(S^ 2)$$ is defined by $$T_ pf(x)=\sum_{S\in S_ p}f(Sx)$$ ($$S$$ acts in a natural way as a rotation, using the correspondence of $$\text{SU}(2)$$ and $$\text{SO}(3)$$). Generalizing a special result of the first part $$(p=5)$$ the authors prove that $$\lambda_ 1(T_ p)\leq 2\sqrt{p}$$ if $$p\equiv 1\pmod 4$$, $$p$$ a prime number.
In the next section a more general scheme for producing very well distributed sequences is presented. One starts with a group $$\Gamma$$ diagonally embedded in $$G_ 1\times\text{SU}(2)$$, where $$G_ 1=\text{PGL}(2,{\mathbb Q}_ p)$$ or $$\text{PSL}(2,{\mathbb R})$$, and for which the projection of $$\Gamma$$ on $$G_ 1$$ is discrete and co-compact. If the elements in $$\Gamma$$ are ordered by a “lattice type” ordering in $$G_ 1$$, the corresponding projections on $$\text{SU}(2)$$ give the desired sequences. In the $$p$$-adic case $$(p<\infty)$$, very good estimates on the spherical cap discrepancy are obtained $$(D\ll (\log N^{2/3})/N^{1/3}).$$
This method is not so successful in the hyperbolic case $$(p=\infty)$$. By a different approach, using techniques from P. D. Lax and R. S. Phillips [J. Funct. Anal. 46, 280–350 (1982; Zbl 0497.30036)] good but quite a bit weaker estimates are obtained. The proofs make essential use of Deligne’s celebrated methods, from his solution of Weil’s conjecture.

##### MSC:
 11F25 Hecke-Petersson operators, differential operators (one variable) 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 11K38 Irregularities of distribution, discrepancy
##### Citations:
Zbl 0619.10052; Zbl 0497.30036
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##### References:
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