  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
${1}/{2}$-rigid transformation $T$ if and only if the complement of $M$ is a
thick set. This result is generalized to ${r}/{(r+1)}$-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.
