The multilinear circle method and a question of Bergelson

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 $T_1,\dots ,T_k\text{\hspace{0.17em}⁣}:X\to X$. Let $P_1,\dots ,P_k\in \mathbb{Z}[\mathrm{n}]$ be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages $$A_{N;X,T_1,\dots ,T_k}^{P_1,\dots ,P_k}(f_1,\dots ,f_k)(x)=\frac{1}{N}\sum _{n=1}^N{f}_1\left(T_1^{P_1(n)}x\right)\cdots f_k\left(T_k^{P_k(n)}x\right),x\in X,$$cas $N\to \mathrm{\infty }$ for any functions $f_1,\dots ,f_k\in L^{\mathrm{\infty }}(X)$. 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.