Pointwise convergence of multiple ergodic averages and strictly ergodic models

By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},\mu, T)$, $d\in\mathbb{N}$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, the averages $\frac{1}{N^2}\sum_{(n,m)\in [0,N-1]^2}f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x)$ converge $\mu$ a.e. Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge a.e. That is, we show that if $(X,\mathcal{X},\mu, T)$ is an ergodic distal system, and $f_1, \ldots, f_d \in L^{\infty}(\mu)$, then multiple ergodic averages $\frac{1}{N}\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x)$ converge $\mu$ a.e.. Joint work with Song Shao and Xiangdong Ye.

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