Polynomial averages and pointwise ergodic theorems on nilpotent groups

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on σ-finite measure spaces. We also establish corresponding maximal inequalities on L^p for 1 < p≤ ∞ and ρ-variational inequalities on L² for 2 < ρ< ∞. This gives an affirmative answer to the Furstenberg–Bergelson–Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.