Spectral multiplicity for powers of weakly mixing automorphisms

We study the behaviour of the maximal spectral multiplicity $\mathfrak m(R^n)$ for the powers of a weakly mixing automorphism $R$. For some particular infinite sets $A$ we show that there exists a weakly mixing rank-one automorphism $R$ such that $\mathfrak m(R^n)=n$ and $\mathfrak m(R^{n+1})=1$ for all positive integers $n\in A$. Moreover, the cardinality $\operatorname{cardm}(R^n)$ of the set of spectral multiplicities for the power $R^n$ is shown to satisfy the conditions $\operatorname{cardm}(R^{n+1})=1$ and $\operatorname{cardm}(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: all powers $R^{n}$ have homogeneous spectra and the set of limit points of the sequence $\{\mathfrak m(R^n)/n:n\in \mathbb N \}$ is infinite.
Bibliography: 17 titles.