Pointwise convergence of bilinear averages for two commuting transformations

Let T and S be commuting measure-preserving transformations of a probability space. We prove that, for f,g ∈ L³, the bilinear averages (1/N)∑ₙ₌₁ᴺ f(Tⁿx)g(Sⁿx) converge almost everywhere and satisfy an L³ᐟ² bound for their full r-variation for every r > 2. In particular, this answers the bounded-input convergence question recorded as Problem 19 in Frantzikinakis’s 2016 survey, without an invertibility assumption.

The main analytic estimate controls point-dependent selections of at most J disjoint increments of smooth averages by C√J at L³ × L³ → L³ᐟ². Its proof refines the Gaussian matrix-energy method of two OpenAI manuscripts, with a bound uniform in a variance ratio. We give complete proofs of the specialized inputs and transfer the estimates to integer averages and commuting systems. Applications include positive pointwise return densities and uniform finite corner-fluctuation bounds.

GitHub repository: eulogica/pointwise-commuting-bilinear-averages