Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912115542982656 |
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| author | O'Keeffe, Maximilian |
| author_facet | O'Keeffe, Maximilian |
| contents | Let $(X,μ)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N T_1^{\lfloor P_1(n) \rfloor} \dotsm T_m^{\lfloor P_m(n) \rfloor} f \] converge pointwise $μ$-almost everywhere as $N \to \infty$ for any $f \in L^p(X)$ with $p>1$, where $P_1, \dotsc, P_m$ are Hardy field functions which are "non-polynomial" and have distinct growth rates. To establish pointwise convergence we will prove a long-variational inequality, which will in turn prove that a maximal inequality holds for our averages.
Additionally, by restricting the class of Hardy field functions to those with the same growth rate as $t^c$ for $c>0$ non-integer, we also prove full variational estimates. We are therefore able to provide quantitative bounds on the rate of convergence of exponential sums of the form \[ \frac{1}{N} \sum_{n=1}^N e(ξ_1 \lfloor n^{c_1} \rfloor + \dotsb + \lfloor n^{c_m} \rfloor) \] where $0<c_1<\dotsb<c_m$ are non-integer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07385 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences O'Keeffe, Maximilian Dynamical Systems Classical Analysis and ODEs Let $(X,μ)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N T_1^{\lfloor P_1(n) \rfloor} \dotsm T_m^{\lfloor P_m(n) \rfloor} f \] converge pointwise $μ$-almost everywhere as $N \to \infty$ for any $f \in L^p(X)$ with $p>1$, where $P_1, \dotsc, P_m$ are Hardy field functions which are "non-polynomial" and have distinct growth rates. To establish pointwise convergence we will prove a long-variational inequality, which will in turn prove that a maximal inequality holds for our averages. Additionally, by restricting the class of Hardy field functions to those with the same growth rate as $t^c$ for $c>0$ non-integer, we also prove full variational estimates. We are therefore able to provide quantitative bounds on the rate of convergence of exponential sums of the form \[ \frac{1}{N} \sum_{n=1}^N e(ξ_1 \lfloor n^{c_1} \rfloor + \dotsb + \lfloor n^{c_m} \rfloor) \] where $0<c_1<\dotsb<c_m$ are non-integer. |
| title | Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences |
| topic | Dynamical Systems Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2411.07385 |