Pointwise ergodic theorems along fractional powers of primes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909422087831552 |
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| author | Bahnson, Erik Daskalakis, Leonidas Dohadwala, Abbas Shah, Ish |
| author_facet | Bahnson, Erik Daskalakis, Leonidas Dohadwala, Abbas Shah, Ish |
| contents | We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences $\lfloor h(p)\rfloor$, where $h$ belongs in a wide class of functions, the so-called $c$-regularly varying functions. We also establish uniform multiparameter oscillation estimates for our ergodic averages and the corresponding multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $L^1$ to not seem entirely out of reach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise ergodic theorems along fractional powers of primes Bahnson, Erik Daskalakis, Leonidas Dohadwala, Abbas Shah, Ish Dynamical Systems Classical Analysis and ODEs Number Theory 37A46, 42B25, 11L20, 11P05 We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences $\lfloor h(p)\rfloor$, where $h$ belongs in a wide class of functions, the so-called $c$-regularly varying functions. We also establish uniform multiparameter oscillation estimates for our ergodic averages and the corresponding multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $L^1$ to not seem entirely out of reach. |
| title | Pointwise ergodic theorems along fractional powers of primes |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory 37A46, 42B25, 11L20, 11P05 |
| url | https://arxiv.org/abs/2412.07055 |