Pointwise convergence of ergodic averages along quadratic bracket polynomials
Leonidas Daskalakis

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 , where is an arbitrary positive rational number with . Namely, we prove that for every such , every measure-preserving system and every , we have that \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n\lfloor n\sqrt{k}\rfloor}x)\quad\text{exists for -a.e. .} \] Notably, our analysis involves a curious implementation of the circle method developed for analyzing exponential sums with phases 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 such a…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematical Dynamics and Fractals · Limits and Structures in Graph Theory · Random Matrices and Applications
