  We study double ergodic averages with respect to two general commuting
transformations and establish a sharp quantitative result on their convergence
in the norm. We approach the problem via real harmonic analysis, using recently
developed methods for bounding multilinear singular integrals with certain
entangled structure. A byproduct of our proof is a bound for a two-dimensional
bilinear square function related to the so-called triangular Hilbert transform.
