Pointwise convergence of some multiple ergodic averages

Abstract We show that for every ergodic system ( X , μ , T 1 , … , T d ) with commuting transformations, the average 1 N d + 1 ∑ 0 ≤ n 1 , … , n d ≤ N − 1 ∑ 0 ≤ n ≤ N − 1 f 1 ( T 1 n ∏ j = 1 d T j n j x ) f 2 ( T 2 n ∏ j = 1 d T j n j x ) ⋯ f d ( T d n ∏ j = 1 d T j n j x ) converges for μ -a.e. x ∈ X as N → ∞ . If X is distal, we prove that the average 1 N ∑ n = 0 N − 1 f 1 ( T 1 n x ) f 2 ( T 2 n x ) ⋯ f d ( T d n x ) converges for μ -a.e. x ∈ X as N → ∞ . We also establish the pointwise convergence of averages along cubical configurations arising from a system with 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.