标题： 沿二次括号多项式的遍历平均值的逐点收敛性 显示英文标题

标题： Pointwise convergence of ergodic averages along quadratic bracket polynomials

摘要： 我们建立了一个逐点收敛结果，该结果针对沿形式为$(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$的轨道的遍历平均值，其中$k$是一个任意的正有理数，满足$\sqrt{k}\not\in\mathbb{Q}$。 即，我们证明对于每个这样的$k$，每个保测系统$(X,\mathcal{B},\mu,T)$和每个$f\in L^{\infty}_{\mu}(X)$，我们都有\[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n\lfloor n\sqrt{k}\rfloor}x)\quad\text{exists for $\mu$-a.e. $x\in X$.} \]显著的是，我们的分析涉及一种有趣的圆法应用，该方法是为分析相位为$(\xi n \lfloor n\sqrt{k}\rfloor)_{1\le n\le N}$的指数和而开发的，这些相位表现出超越有理数且分母较小的算术障碍，其基础是 Green 和 Tao 关于多项式轨道在 nilmanifolds 上的定量行为的结果。对于情况$k=2$，这种圆法首次被用于解决相应的 Waring 类型问题，由 Neale 提出，他们的工作构成了我们考虑的出发点。 显示英文摘要

摘要： 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 $(X,\mathcal{B},\mu,T)$ and every $f\in L^{\infty}_{\mu}(X)$, we have that \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n\lfloor n\sqrt{k}\rfloor}x)\quad\text{exists for $\mu$-a.e. $x\in X$.} \] Notably, our analysis involves a curious implementation of the circle method developed for analyzing exponential sums with phases $(\xi n \lfloor n\sqrt{k}\rfloor)_{1\le n\le N}$ 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.