Pointwise recurrence for commuting measure preserving transformations

Let $(X,\mathcal{A},\mu )$ be a probability measure space and let $T_i,$ $1\le i\le H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages $$\frac{1}{N}\sum _{n=1}^N{f}_1(T_1^{n}x)f_2(T_2^{n}x)\cdots f_H(T_H^{n}x)$$ converge a.e. for every function $f_i\in L^{\mathrm{\infty }}(\mu )$ .\ As a consequence if $T_i = T^i$for$1\leq i \leq H$where$T$ is an invertible measure preserving transformation on $(X,\mathcal{A},\mu )$ then the averages $$\frac{1}{N}\sum _{n=1}^N{f}_1(T^{n}x)f_2(T^{2n}x)\mathrm{.}\mathrm{.}\mathrm{.}{f}_H(T^{Hn}x)$$ converge a.e. This solves a long open question on the pointwise convergence of nonconventional ergodic averages. For $H=2$ it provides another proof of J. Bourgain’s a.e. double recurrence theorem.