Cyclic extensions of odometer transformations and spectral disjointness (Q1100302)

Let \({\mathbb{Z}}_ r\) be the set of r-adic integers. The map \(T_ r:\) \({\mathbb{Z}}_ r\to {\mathbb{Z}}_ r\) defined by \(T_ r\omega =\omega +1\) together with Haar measure gives a dynamical system which is called the odometer to base r. Let \(\xi \in S^ 1\) and denote by G(\(\xi)\) the smallest closed subgroup of \(S^ 1\) which contains all powers of \(\xi\). Furthermore let \(\psi\) : \({\mathbb{Z}}_ r\to {\mathbb{Z}}\) be the map defined by \(\psi (-1)=0\) and \[ \psi (\omega)=\lim_{N\to \infty}(\sum^{N}_{n=0}(\omega +1)_ n-\sum^{N}_{n=0}\omega_ n). \] Then the transformation \(T(\omega,g)=(\omega +1,g\xi^{\psi (\omega)})\) induces a dynamical system (the cyclic extension of \(T_ r)\) on \({\mathbb{Z}}_ r\times G.\) It is shown that T has discrete spectrum if and only if \(\xi^{r-1}=1\). Furthermore under some additional assumptions it is shown that two cyclic extensions belonging to different pairs (r,\(\xi)\) and (s,\(\eta)\) are spectrally disjoint. A corollary of further investigations includes the following result of \textit{W. Schmidt} [Pac. J. Math. 10, 661-672 (1960; Zbl 0093.054)]: Let r and s be multiplicatively independent numbers; then the set of numbers which are normal to base r and not normal to base s has the cardinality of continuum.