Pointwise ergodic theorems for non-conventional bilinear polynomial averages

• 成果类型: Article
• 署名作者: Krause, Ben; Mirek, Mariusz; Tao, Terence
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.195.3.4
• 发表日期: 2022
• 页码: 997-1109
• 关键词: jump inequalities; hilbert-transforms; harmonic-analysis; CONVERGENCE; Boundedness; UNIFORMITY; SEQUENCES; subsets; PROOF

摘要

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages AN(f,g)(x) := 1/N Sigma(N)(n=1) f(T(n)x)g(T(P(n))x) as N -> infinity, where T: X -> X is a measure-preserving transformation of a sigma-finite measure space (X, mu), P(n) is an element of Z[n] is a polynomial of degree d >= 2, and f is an element of L-P1 (X), g is an element of L-P2 (X) for some p1 ,p 2 > 1 with 1/p1 + 1/p2 <= 1. We also establish an r-variational inequality for these averages (at lacunary scales) in the optimal range r > 2. We are also able to break duality by handling some ranges of exponents p1, p2 with 1/p1 + 1/p2 > 1, at the cost of increasing r slightly. This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg-Weiss averages (with P(n) = n(2)), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. This also gives a contribution to the Furstenberg-Bergelson-Leibman conjecture. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers A(z) also plays a role.