2024/11/14 by Dariusz Kosz, Kosz, Dariusz, Mariusz Mirek +5 · 4 citations
Computer Science · Mathematics · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2411.09478
openalex publication_date 2024/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k∈ \mathbb Z+ and (X, \mathcal B(X), μ) be a probability space equipped with a family of commuting invertible measure-preserving transformations T1,…, Tk \colon X→ X. Let P1,…, Pk∈\mathbb Z[\rm n] be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages AN; X, T1,…, TkP1,…, Pk(f1,…, fk)(x) = (1)/(N)∑n=1Nf1(T1P1(n)x)⋯ fk(TkPk(n)x), x∈ X,cas N→∞ for any functions f1, …, fk∈ L∞(X). Besides a couple of results in the bilinear setting (when k=2 and then only for single transformations), this is the first pointwise result for general polynomial multilinear ergodic averages in arbitrary measure-preserving systems. This answers a question of Bergelson from 1996 in the affirmative for any polynomials with distinct degrees, and makes progress on the Furstenberg-Bergelson-Leibman conjecture. In this paper, we build a versatile multilinear circle method by developing the Ionescu-Wainger multiplier theorem for the set of canonical fractions, which gives a positive answer to a question of Ionescu and Wainger from 2005. We also establish sharp multilinear Lp-improving bounds and an inverse theorem in higher order Fourier analysis for averages over polynomial corner configurations, which we use to establish a multilinear analogue of Weyl's inequality and its real counterpart, a Sobolev smoothing inequality.