Pointwise almost everywhere convergence of multiple ergodic averages

Establish pointwise almost everywhere convergence for multiple ergodic averages A_N(f_1,...,f_ℓ)= (1/N)∑_{n=1}^N T_1^{a_1(n)}f_1⋯T_ℓ^{a_ℓ(n)}f_ℓ along integer polynomial iterates a_1,...,a_ℓ for commuting invertible measure-preserving transformations T_1,...,T_ℓ on standard probability spaces.

Background

The survey contrasts the well-developed theory of norm convergence with the much less understood pointwise almost everywhere convergence for multiple ergodic averages.

Despite recent breakthroughs in special cases, a general pointwise theory for polynomial (and other) multiple averages remains elusive.