Pointwise convergence of some multiple ergodic averages

We show that for every ergodic system (X, mu, T-1 , . . . , T-d) with commuting transformations, the average 1/Nd+1 Sigma(0 <= n1 ,..., nd <= N-1 )Sigma(0 <= n <= N-1 ) f(1)(T-1(n) Pi(d)(j=1) T(j)(nj)x) f(2)(T-2(n) Pi(d)(j=1)T(j)(ni)x) . . . f(d)(T-d(n) Pi(d)(j=1)T(j)(nj)x) converges for mu-a.e. x Sigma X as N -> infinity. If X is distal, we prove that the average 1/N Sigma(N-1)(n=0) f(1) (T(1)(n)x) f(2) (T(2)(n)x) . . . f(d)(T(d)(n)x) converges for mu-a.e. x is an element of X as N -> infinity. 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. (C) 2018 Elsevier Inc. All rights reserved.