On weak product recurrence and synchronization of return times

The paper is devoted to a study of product recurrence. First, we prove that notions of ℱps-PR and ℱpubd-PR are exactly the same as product recurrence, completing that way results of [P. Dong, S. Shao, X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. Math. 188 (1) (2012) 463-507], and consequently, extending the characterization of return times of distal points which originated from the 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 ℱs-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).