Quantitative Convergence for Sparse Ergodic Averages in $L^1$
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917323416272896 |
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| author | Krause, Ben Sun, Yu-Chen |
| author_facet | Krause, Ben Sun, Yu-Chen |
| contents | We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E} X_j = n^{-α}$, and we restrict to $1 < c < 7/6, \ 0 < α< 1/2$.
Then (almost surely) for any measure-preserving system, $(X,μ,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $μ$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain.
This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c = \frac{1001}{1000}, \ \frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Convergence for Sparse Ergodic Averages in $L^1$ Krause, Ben Sun, Yu-Chen Dynamical Systems Classical Analysis and ODEs Number Theory Probability We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E} X_j = n^{-α}$, and we restrict to $1 < c < 7/6, \ 0 < α< 1/2$. Then (almost surely) for any measure-preserving system, $(X,μ,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $μ$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c = \frac{1001}{1000}, \ \frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting. |
| title | Quantitative Convergence for Sparse Ergodic Averages in $L^1$ |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory Probability |
| url | https://arxiv.org/abs/2504.12510 |