Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918411413487616 |
|---|---|
| author | Fornal, Jan Krause, Ben |
| author_facet | Fornal, Jan Krause, Ben |
| contents | Let $(X,μ)$ be a probability space equipped with an invertible, measure-preserving transformation $T\colon X \to X$. We exhibit a wide class of weights $w$ so that whenever $f,g \in L^{\infty}(X)$, the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} w(n)\, T^{an}f \cdot T^{bn}g, \qquad a,b \in \mathbb{Z} \] converge $μ$-almost surely. This class encompasses the von Mangoldt function, resolving Problem 12 from Frantzikinakis' survey on open problems in ergodic theory, the divisor function, the sum-of-two-squares representation function, etc., as well as their restrictions to lower-density Piatetski-Shapiro sequences of the form $\{\lfloor k^{c}\rfloor : k \in \mathbb{N}\}$, $1 \leq c < 7/6$.
Our methods combine combinatorial number theory and higher-order Fourier analysis with classical Fourier-analytic/martingale-based methods; the role of $U^{3}$ analysis is particularly significant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22203 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory Fornal, Jan Krause, Ben Dynamical Systems Classical Analysis and ODEs Number Theory Let $(X,μ)$ be a probability space equipped with an invertible, measure-preserving transformation $T\colon X \to X$. We exhibit a wide class of weights $w$ so that whenever $f,g \in L^{\infty}(X)$, the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} w(n)\, T^{an}f \cdot T^{bn}g, \qquad a,b \in \mathbb{Z} \] converge $μ$-almost surely. This class encompasses the von Mangoldt function, resolving Problem 12 from Frantzikinakis' survey on open problems in ergodic theory, the divisor function, the sum-of-two-squares representation function, etc., as well as their restrictions to lower-density Piatetski-Shapiro sequences of the form $\{\lfloor k^{c}\rfloor : k \in \mathbb{N}\}$, $1 \leq c < 7/6$. Our methods combine combinatorial number theory and higher-order Fourier analysis with classical Fourier-analytic/martingale-based methods; the role of $U^{3}$ analysis is particularly significant. |
| title | Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2603.22203 |