A double return times theorem (Q1717606)

\textit{J. Bourgain}'s bilinear pointwise ergodic theorem, one of the hardest known convergence results for multiple ergodic averages, has been extended by \textit{I. Assani} et al. [Ergodic Theory Dyn. Syst. 36, No. 4, 1037--1066 (2016; Zbl 1353.37006)] to a Wiener-Wintner type result by relatively simple means, using Bourgain's result as a black box [\textit{J. Bourgain}, J. Reine Angew. Math. 404, 140--161 (1990; Zbl 0685.28008)]. The current paper strengthens the latter result further to a double return times theorem, similarly assuming Bourgain's result as a black box.