The multilinear circle method and a question of Bergelson
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909909127266304 |
|---|---|
| author | Kosz, Dariusz Mirek, Mariusz Peluse, Sarah Wan, Renhui Wright, James |
| author_facet | Kosz, Dariusz Mirek, Mariusz Peluse, Sarah Wan, Renhui Wright, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09478 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The multilinear circle method and a question of Bergelson Kosz, Dariusz Mirek, Mariusz Peluse, Sarah Wan, Renhui Wright, James Dynamical Systems Classical Analysis and ODEs Number Theory 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. |
| title | The multilinear circle method and a question of Bergelson |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2411.09478 |