Weakly tame systems, their characterizations and application.

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 introduce a notion of 'weak-tameness', which is a measure-theoretic version of a notion of `tameness' introduced by E. Glasner, based on the work of A. K\"ohler [A. K\"ohler introduced this notion and call such systems "regular".], and characterize such topological dynamical systems as systems for which every invariant measure has a discrete spectrum. Using the work of M. Talagrand, we also characterize weakly tame as well as tame systems in terms of the notion of 'witness of irregularity' which is based on up-crossings. Then we establish that 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. Thus Sarnak's M\"obius orthogonality conjecture holds for this flow and as a consequence, we obtain an improvement of Motohashi-Ramachandra 1976's theorem on the Mertens function in short intervals. We further improve Motohashi-Ramachandra's bound to $1/2$ under Chowla conjecture.