Question

Let $\operatorname{frac}(x) = x - \lfloor x \rfloor$ be the fractional part of $x$. Then, for $\alpha$ irrational, $\operatorname{frac}(n \alpha)$, $n=1,2,\ldots$, distributes randomly in $[0,1)$, which can be viewed as the circle $\mathbb{S}^1$. Weyl's Equidistribution Theorem establishes the uniformity of the distribution.
          
           _{$\operatorname{frac}(n \pi)$ for $n=1,2,\ldots,100$.}

Is there an analog to $\operatorname{frac}(n \alpha)$ for $\mathbb{S}^2$? Is there a function $f(n,x)$ with the behavior that, for some point $x \in \mathbb{S}^2$, $f(n,x)$ for $n=1,2,\ldots$ fills $\mathbb{S}^2$ randomly and uniformly?

A pointer would suffice if this is well known. Thanks!

Answer 1

If you take two random elements in $SO(3),$ (this is sort of like irrationals in $SO(2)$) the group they generate is both free and equidistributed in $SO(3),$ so the orbit of the north pole (or any other point you favor) will be equidistributed in $\mathbb{S}^2.$ You may complain that a free group will need two indices, but since free groups are orderable, you can make the two indices into one.

Answer 2

Kuipers and Niederreiter, Uniform Distribution of Sequences, give these references for questions of uniform distribution of sequences on a sphere:

V I Arnol'd, A L Krylov, Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex region, Dokl Akad Nauk SSSR 148 (1963) 9-12; English translation, Soviet Math Dokl 4 (1963) 1-5.

P Gerl, Gleichverteilung auf der Kugel, Arch der Math 24 (1973) 203-207.

Those of us whose familiarity with kugel extends only to the side dish commonly served on the Shabbes and other holidays (http://en.wikipedia.org/wiki/Kugel) may find some amusement in the title of Gerl's paper.