Pointwise convergence of ergodic averages along quadratic bracket polynomials

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 $\sqrt{k}\in ̸\mathbb{Q}$. Namely, we prove that for every such $k$, every measure-preserving system $(X,\mathcal{B},\mu ,T)$ and every $f\in L_{\mu }^{\mathrm{\infty }}(X)$, we have that $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{n=1}^{N}f(T^{n\lfloor n\sqrt{k}\rfloor }x)\text{exists for }\mu \text{-a.e. }x\in X\text{.}$$ 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.