Gutman, Yonatan et al., Almost sure convergence of the multiple ergodic average for certain weakly mixing systems, Acta mathematica Sinica. English series (Print) , 34(1), 2018, pp. 79-90
The family of pairwise independently determined (PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages 1N.n=0N.1f1(Tnx)...fd(Tdnx),N.. almost surely converge.