On Gruber's problem concerning uniform distribution on the sphere (Q651395)

The author studies a problem by Peter Gruber that runs as follows: Suppose that a sequence \((u_j)_{j\geq 1}\) on \(d\)-dimensional the unit sphere satisfies \[ \lim_{N\to\infty} \frac 1N \sum_{j=1}^N \chi_{\rho\, C}(u_j) = \mu(C) \] for a given cap \(C\) and all rotations \(\rho\). Is it true that the sequence \((u_j)_{j\geq 1}\) is then uniformly distributed on the sphere. (\(\chi_M\) denotes the characteristic function of \(M\) and \(\mu\) the normalized Haar measure.) In the 1-dimensional case this is true if and only if \(\mu(C)\) is irrational. In the higher dimensional case the author provides a partial answer. He shows that there exists a countable and dense set \(D \subseteq [0,1]\) such that a sequence \((u_j)_{j\geq 1}\) need not be uniformly distributed when \(\mu(C) \in D\) and the above limiting distribution holds for all rotations \(\rho\). The proof is a consequence of a result of \textit{P.\ Ungar} [``Freak theorem about functions on a sphere'', J. Lond. Math. Soc. 29, 100--103 (1954; Zbl 0058.28403)].