An improved characterization of normal sets and some counter-examples (Q1288499)

The author gives a characterization of normal sets which strengthens a characterization given by G. Rauzy. He shows that the class of normal sets is closed under finite unions and countable intersections. Further he shows that there exists sequences \(U\) and \(V\) of real numbers such that \(\alpha U+\beta V\) is uniformly distributed mod~1 if and only if one of the real numbers \(\alpha,\beta\) vanishes. He also constructs a sequence \(U\) of integers which is uniformly distributed in \(\mathbb{Z}\) but such that for any real \(\alpha\), \(\alpha U\) is not uniformly distributed mod~1.