Pointwise convergence of ergodic averages along quadratic bracket polynomials

TL;DR

This paper proves that ergodic averages along quadratic bracket polynomial orbits converge pointwise for all measure-preserving systems and bounded functions, extending previous methods with novel circle method techniques and nilmanifold analysis.

Contribution

It introduces a new approach combining the circle method and nilmanifold techniques to establish pointwise convergence for ergodic averages along quadratic bracket polynomial orbits.

Findings

Pointwise convergence holds for all such orbits and functions.

The analysis involves advanced exponential sum estimates.

Extends previous results using novel circle method applications.

Abstract

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.