Norm convergence of fractional-polynomial averages on nilpotent (and even commuting) systems

Establish L^2(μ) convergence of (1/N)∑_{n=1}^N T_1^{a_1(n)}f_1⋯T_ℓ^{a_ℓ(n)}f_ℓ when a_1,...,a_ℓ are fractional polynomials and T_1,...,T_ℓ generate a nilpotent group of measure-preserving transformations; resolve the problem even for commuting transformations.

Background

Walsh’s theorem gives norm convergence for polynomial iterates on nilpotent systems, but fractional powers fall outside its scope.

The authors emphasize that even in the commuting case the problem is unsolved.