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We show that for every ergodic system $(X,\mu,T_1,\ldots,T_d)$ with commutingtransformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d\leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n\prod_{j=1}^d T_j^{n_j}x)\cdots f_d(T_d^n \prod_{j=1}^d T_j^{n_j}x). \]converges for $\mu$-a.e. $x\in X$ as $N\to\infty$. If $X$ is distal, we provethat the average \[\frac{1}{N}\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\cdotsf_d(T_d^nx) \] converges for $\mu$-a.e. $x\in X$ as $N\to\infty$. We alsoestablish the pointwise convergence of averages along cubical configurationsarising from a system commuting transformations. Our methods combine theexistence of sated and magic extensions introduced by Austin and Hostrespectively with ideas on topological models by Huang, Shao and Ye.

Pointwise convergence of some multiple ergodic averages