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Pointwise convergence of ergodic averages along quadratic bracket polynomials

Leonidas Daskalakis

TL;DR

The authors prove almost everywhere convergence of ergodic averages along the sparse quadratic-bracket orbit $\{n\lfloor n\sqrt{k}\rfloor\}$ for rational $k>0$ with $\sqrt{k}$ irrational, by developing a specialized circle-method framework. They decompose the exponential sums into major-arc and minor-arc components, deriving a precise major-arc factorization $m_{N;k}(\xi)\approx \mathrm{G}_{k}(a,b,q)\mathrm{F}(b/(2q))V_{N;k}(t)$ and establishing decay bounds on both arcs. Central to their analysis are the Green–Tao quantitative Leibman theorems on polynomial orbits on nilmanifolds (applied on the 3D Heisenberg nilmanifold) and a sequence of 2-oscillation estimates that yield almost-everywhere convergence via lacunary scales and Bourgain’s logarithmic lemma. By Calderón transference, these results extend to general measure-preserving systems, yielding a new class of pointwise ergodic theorems for generalized bracket-polynomial orbits. The work also outlines future directions for generalized bracket polynomials and the development of an L^p theory for such averages, highlighting connections to state-of-the-art quantitative ergodic results.

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 $(X,\mathcal{B},μ,T)$ 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.

Pointwise convergence of ergodic averages along quadratic bracket polynomials

TL;DR

The authors prove almost everywhere convergence of ergodic averages along the sparse quadratic-bracket orbit for rational with irrational, by developing a specialized circle-method framework. They decompose the exponential sums into major-arc and minor-arc components, deriving a precise major-arc factorization and establishing decay bounds on both arcs. Central to their analysis are the Green–Tao quantitative Leibman theorems on polynomial orbits on nilmanifolds (applied on the 3D Heisenberg nilmanifold) and a sequence of 2-oscillation estimates that yield almost-everywhere convergence via lacunary scales and Bourgain’s logarithmic lemma. By Calderón transference, these results extend to general measure-preserving systems, yielding a new class of pointwise ergodic theorems for generalized bracket-polynomial orbits. The work also outlines future directions for generalized bracket polynomials and the development of an L^p theory for such averages, highlighting connections to state-of-the-art quantitative ergodic results.

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 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 circle method was firstly employed for addressing the corresponding Waring-type problem by Neale, and their work constitutes the departure point of our considerations.

Paper Structure

This paper contains 29 sections, 31 theorems, 327 equations.

Table of Contents

  1. Introduction
  2. Introductory remarks
  3. Statement of results
  4. Strategy
  5. Minor arc estimates
  6. Major arcs estimates
  7. Establishing oscillation estimates
  8. Further directions
  9. Generalized (or bracket) polynomials.
  10. The $L^p$-theory.
  11. State-of-the-art quantitative ergodic theorems
  12. Notation
  13. Setting up the circle method
  14. Major arc estimates
  15. Major arc approximation
  16. ...and 14 more sections

Key Result

Theorem 1.4

Assume $k\in\mathbb{Q}_{>0}$ is such that $\sqrt{k}\not\in\mathbb{Q}$. Let $(X,\mathcal{B},\mu)$ be a probability space and $T\colon X\to X$ an invertible $\mu$-invariant transformation. Then for every $f\in L^{\infty}_{\mu}(X)$ we have that

Theorems & Definitions (57)

  • Theorem 1.4: Pointwise ergodic theorem along $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$
  • Theorem 1.7: Oscillation estimates along lacunary scales on $L^2_{\mu}(X)$
  • Theorem 1.10
  • Lemma 3.4: Unique representation for the major arc centers
  • Lemma 3.6: General separation condition
  • proof
  • Lemma 3.9: Separation condition for the major arc centers
  • proof
  • Proposition 4.1: Major arc approximation
  • Lemma 4.5
  • ...and 47 more