  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.
