The multilinear circle method and a question of Bergelson

TL;DR

The paper proves pointwise almost everywhere convergence for multilinear polynomial ergodic averages $A_{N; X, \mathcal{T}}^{\mathcal{P}}(f_1,\dots,f_k)$ in arbitrary measure-preserving systems for polynomials with pairwise distinct degrees, advancing Bergelson's question and the Furstenberg–Bergelson–Leibman conjecture. The authors develop a robust multilinear circle method built around an Ionescu–Wainger multiplier theorem for canonical fractions, complemented by inverse theorems, multilinear $\ell^p$-improving bounds, and multilinear Weyl/Sobolev smoothing inequalities to control minor and major arc contributions. A key contribution is the reduction to the integer shift system via Calderón transference, followed by a detailed treatment of major/minor arcs and the introduction of a multilinear $\ell^p$-improving framework that yields polynomial decay bounds without $\varepsilon$-loss. The results yield norm and pointwise convergence, maximal and variational estimates, and progress toward the Furstenberg–Bergelson–Leibman conjecture in the commutative setting, with potential extensions via the multilinear circle method to broader polynomial configurations. These methods provide a flexible, p-adic-free approach that may illuminate several longstanding open questions in higher-order ergodic theory and additive combinatorics.

Abstract

Let $k\in \mathbb Z_+$ and $(X, \mathcal B(X), μ)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots, T_k \colon X\to X$. Let $P_1,\ldots, P_k\in\mathbb Z[\rm n]$ be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages \[A_{N; X, T_1,\ldots, T_k}^{P_1,\ldots, P_k}(f_1,\ldots, f_k)(x) = \frac{1}{N}\sum_{n=1}^Nf_1\big(T_1^{P_1(n)}x\big)\cdots f_k\big(T_k^{P_k(n)}x\big), \qquad x\in X, \]cas $N\to\infty$ for any functions $f_1, \ldots, f_k\in L^{\infty}(X)$. 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 $L^p$-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.

Theorems & Definitions (110)

• Theorem 1.3
• Theorem 1.4: The Bergelson--Leibman polynomial Szemerédi theorem, 1996 BL1
• Theorem 1.6: Frantzikinakis--Kuca, 2022
• Corollary 1.8
• Theorem 1.12
• Conjecture 1.15: The Furstenberg--Bergelson--Leibman conjecture
• Example 1.18
• Theorem 1.21
• Theorem 1.39: Peluse, 2020
• Proposition 3.1