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تسجيل مستخدم جديدAlmost sure convergence of the multiple ergodic average for certain weakly mixing systems
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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 begin{equation*} frac 1 Nsum_{n=0}^{N-1}f_1(T^nx)cdots f_d(T^{dn}x), quad Nto infty, end{equation*} almost surely converge.
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