Pointwise convergence of polynomial multiple ergodic averages along the primes
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918029107920896 |
|---|---|
| author | Wan, Renhui |
| author_facet | Wan, Renhui |
| contents | We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages $$\frac{1}{N} \sum_{n=1}^N \La(n) f_1(T^{P_1(n)} x)\cdots f_k(T^{P_k(n)} x)$$ as $N\to \infty$, where $\La$ is the von Mangoldt function, $T \colon X \to X$ is an invertible measure-preserving transformation of a probability space $(X,ν)$, $P_1,\ldots, P_k$ are polynomials with integer coefficients and distinct degrees, and $f_1,\ldots,f_k\in L^\infty(X)$. This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees.
Building on the foundational work of Krause--Mirek--Tao (Ann. of Math., 2022), Kosz--Mirek--Peluse--Wright (arXiv: 2411.09478, 2024), and Krause--Mousavi--Tao--Teräväinen (arXiv: 2409.10510, 2024), we develop a multilinear circle method for
von Mangoldt-weighted (equivalently, prime-weighted) averages. This method combines harmonic analysis techniques across multiple groups with the newest inverse theorem from additive combinatorics. In particular, the principal innovations of this framework include:
(i) an inverse theorem and a Weyl-type inequality for multilinear Cramér-weighted averages; (ii) a multilinear Rademacher-Menshov inequality; and (iii) an arithmetic multilinear estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15549 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence of polynomial multiple ergodic averages along the primes Wan, Renhui Dynamical Systems Classical Analysis and ODEs Number Theory We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages $$\frac{1}{N} \sum_{n=1}^N \La(n) f_1(T^{P_1(n)} x)\cdots f_k(T^{P_k(n)} x)$$ as $N\to \infty$, where $\La$ is the von Mangoldt function, $T \colon X \to X$ is an invertible measure-preserving transformation of a probability space $(X,ν)$, $P_1,\ldots, P_k$ are polynomials with integer coefficients and distinct degrees, and $f_1,\ldots,f_k\in L^\infty(X)$. This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. Building on the foundational work of Krause--Mirek--Tao (Ann. of Math., 2022), Kosz--Mirek--Peluse--Wright (arXiv: 2411.09478, 2024), and Krause--Mousavi--Tao--Teräväinen (arXiv: 2409.10510, 2024), we develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages. This method combines harmonic analysis techniques across multiple groups with the newest inverse theorem from additive combinatorics. In particular, the principal innovations of this framework include: (i) an inverse theorem and a Weyl-type inequality for multilinear Cramér-weighted averages; (ii) a multilinear Rademacher-Menshov inequality; and (iii) an arithmetic multilinear estimate. |
| title | Pointwise convergence of polynomial multiple ergodic averages along the primes |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2505.15549 |