Spectral multiplicities for ergodic flows

Abstract

Let $E$ be a subset of positive integers such that $E\cap \{1,2\}\ne \varnothing$. A weakly mixing finite measure preserving flow $T=(T_t{)}_{t\in \mathbb{R}}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in \mathbb{R}$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.