  The paper is devoted to study of product recurrence. First, we prove that
notions of $\F_{ps}-PR$ and $\F_{pubd}-PR$ are exactly the same as product
recurrence, completing that way results of [P. Dong, S. Shao and X. Ye,
\emph{Product recurrent properties, disjointness and weak disjointness}, Israel
J. Math.], and consequently, extending the characterization of return times of
distal points which originated from works of Furstenberg. We also study the
structure of the set of return times of weakly mixing sets. As a consequence,
we obtain new sufficient conditions for $\F_{s}-PR$ and also find a short proof
that weakly mixing systems are disjoint with all minimal distal systems (in
particular, our proof does not involve Furstenberg's structure theorem of
minimal distal systems).
