Mixing sequences for non-mixing transformations and group actions

We establish that there are non-mixing maps that are mixing on appropriate sequences including sequences $(s_i)$ which satisfy the Rajchman dissociated property. Our examples are based on the staircase rank one construction, $M$-towers constructions and the Gaussian transformations. As a consequence, we obtain there are non-mixing maps which are mixing along the squares. We further prove that a sequence $M=(m_n)$ is a mixing sequence for some weak mixing $1\mathrm{/}2$-rigid transformation $T$ if and only if the complement of $M$ is a thick set. This result is generalized to $r\mathrm{/}(r+1)$-rigid transformations for $r\in \mathbb{N}$. Moreover, by applying Host-Parreau characterization of the set of continuity from Harmonic Analysis, we extend our results to the infinite countable abelian group actions.