Abstract
Let $k \in \mathbb{Z}_+$ and $(X,\mathcal{B}(X),\mu)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots,T_k: X\to X$. Let $P_1,\ldots,P_k\in \mathbb{Z}[n]$ be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages
\[A_{N;X,T_1,\ldots,T+k}^{P_1,\ldots P_k} (f_1,\ldots,f_k)(x) := \frac{1}{N} \sum_{n=1}^N f_1\bigl(T_1^{P_1(n)}x\bigr) \cdots f_k\bigl(T_k^{P_k(n)}x\bigr), \quad x\in X,\]
as $N\to \infty$ for any functions $f_1,\ldots,f_k \in L^\infty(K)$. 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 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 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 estimate.
