Pointwise convergence of some multiple ergodic averages (1609.02529v1)
Abstract: We show that for every ergodic system $(X,\mu,T_1,\ldots,T_d)$ with commuting transformations, 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_1n \prod_{j=1}d T_j{n_j}x)f_2(T_2n \prod_{j=1}d T_j{n_j}x)\cdots f_d(T_dn \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 prove that the average [\frac{1}{N}\sum_{i=0}{N} f_1(T_1nx)f_2(T_2nx)\cdots f_d(T_dnx) ] converges for $\mu$-a.e. $x\in X$ as $N\to\infty$. We also establish the pointwise convergence of averages along cubical configurations arising from a system commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.