Veech systems and spectral characterization of tameness with applications to Number Theory

We explore the notion of discrete spectrum and its various characterizations for ergodic measure preserving actions of an amenable group on a compact metric space. We further present a spectral characterization of tameness and we establish that the strong Veech systems are tame. In particular, for any amenable group $T$ the flow on the orbit closure of the translates of a `Veech function' $f\in \mathbb{K}(T)$ is tame. As a consequence, we obtain an improvement of Motohashi-Ramachandra 1976's theorem on the Mertens function in short interval, by establishing that M\"{o}bius orthogonality conjecture of Sarnak holds for those systems.