Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials

2026/07/31 by Andreas Koutsogiannis, Borys Kuca, Wenbo Sun
Mathematics · #math.DS #math.CO #msc:37A30 #msc:11B30 #msc:28D05 #msc:37A44

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arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

We investigate manifestations of the Nilpotent Heuristic, which posits that recurrence and convergence phenomena known for measure-preserving ℤ^D-systems extend to nilpotent group actions. Our main results establish seminorm estimates and limiting formulas for multiple ergodic averages arising from actions of 2-step nilpotent groups. In particular, if T₁,…,T_ℓ are totally ergodic and generate a 2-step nilpotent group, then lim_N→∞(1)/(N)∑_n=1^N T₁ⁿ f₁ ⋯ T_ℓ^{n^ℓ}f_ℓ = ∏_j=1^ℓ∫ f_j dμ in the L² norm for all bounded functions f₁,…,f_ℓ; the same holds for any distinct-degree polynomial iterates. We also obtain popular-common-difference versions of the polynomial Szemeédi theorem in the same setting. In a different direction, our approach allows us to completely resolve the joint ergodicity conjecture for multidimensional polynomials and \mathbb Z^D-systems; we also present an example showing that, surprisingly enough, the 2-step nilpotent analog fails. We conclude with many open problems concerning joint ergodicity, seminorm estimates, and the structure theory of nilpotent systems.