Pointwise convergence of ergodic averages along quadratic bracket polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911243613241344 |
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| author | Daskalakis, Leonidas |
| author_facet | Daskalakis, Leonidas |
| contents | We establish a pointwise convergence result for ergodic averages modeled along orbits of the form $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$, where $k$ is an arbitrary positive rational number with $\sqrt{k}\not\in\mathbb{Q}$. Namely, we prove that for every such $k$, every measure-preserving system $(X,\mathcal{B},μ,T)$ and every $f\in L^{\infty}_μ(X)$, we have that \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n\lfloor n\sqrt{k}\rfloor}x)\quad\text{exists for $μ$-a.e. $x\in X$.} \] Notably, our analysis involves a curious implementation of the circle method developed for analyzing exponential sums with phases $(ξn \lfloor n\sqrt{k}\rfloor)_{1\le n\le N}$ exhibiting arithmetical obstructions beyond rationals with small denominators, and is based on the Green and Tao's result on the quantitative behaviour of polynomial orbits on nilmanifolds. For the case $k=2$ such a circle method was firstly employed for addressing the corresponding Waring-type problem by Neale, and their work constitutes the departure point of our considerations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_27590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence of ergodic averages along quadratic bracket polynomials Daskalakis, Leonidas Dynamical Systems Classical Analysis and ODEs Number Theory 37A46, 42B25, 11L07 We establish a pointwise convergence result for ergodic averages modeled along orbits of the form $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$, where $k$ is an arbitrary positive rational number with $\sqrt{k}\not\in\mathbb{Q}$. Namely, we prove that for every such $k$, every measure-preserving system $(X,\mathcal{B},μ,T)$ and every $f\in L^{\infty}_μ(X)$, we have that \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n\lfloor n\sqrt{k}\rfloor}x)\quad\text{exists for $μ$-a.e. $x\in X$.} \] Notably, our analysis involves a curious implementation of the circle method developed for analyzing exponential sums with phases $(ξn \lfloor n\sqrt{k}\rfloor)_{1\le n\le N}$ exhibiting arithmetical obstructions beyond rationals with small denominators, and is based on the Green and Tao's result on the quantitative behaviour of polynomial orbits on nilmanifolds. For the case $k=2$ such a circle method was firstly employed for addressing the corresponding Waring-type problem by Neale, and their work constitutes the departure point of our considerations. |
| title | Pointwise convergence of ergodic averages along quadratic bracket polynomials |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory 37A46, 42B25, 11L07 |
| url | https://arxiv.org/abs/2510.27590 |