Hecke operators and distributing points on \(S^2\). II.

*(English)*Zbl 0648.10034In 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.

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.

Reviewer: Harald Rindler (Wien)

##### 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 |

##### Keywords:

lattice method; evenly distributed sequences on the sphere; Hecke operators; upper bound; eigenvalue; rotation; very well distributed sequences; spherical cap discrepancy
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\textit{A. Lubotzky} et al., Commun. Pure Appl. Math. 40, No. 4, 401--420 (1987; Zbl 0648.10034)

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