Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative Convergence for Sparse Ergodic Averages in $L^1$

Ben Krause, Yu-Chen Sun

TL;DR

This work addresses the challenge of ergodic convergence along sparse times at the $L^1$ endpoint. It develops a unified Calderón-Zygmund framework that treats both deterministic sparse sequences $a_n=\lfloor n^c\rfloor$ (with $1<c<\frac{8}{7}$) and random hitting-time sequences (with $0<\alpha<\frac{1}{2}$), proving almost-sure convergence of $\frac{1}{N}\sum_{n\le N} T^{a_n} f$ for all $f\in L^1(X)$. The authors introduce quantitative oscillation-variation measures (jump-counting, $\mathcal{V}^r$, and $\mathcal{O}_{\{M_j\}}$) to obtain explicit convergence rates, extending Bourgain's technique and improving prior thresholds (e.g., up to $c<\frac{8}{7}$ in the deterministic case and the random setting). The results yield both universal $L^1$-endpoint convergence and quantitative rate information, which enhances understanding of sparsity in ergodic averages and provides a versatile method for analyzing convergence across deterministic and probabilistic constructions with potential applications to sparse ergodic problems and related harmonic-analysis frameworks.

Abstract

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E} X_j = n^{-α}$, and we restrict to $1 < c < 8/7, \ 0 < α< 1/2$. Then (almost surely) for any measure-preserving system, $(X,μ,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $μ$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c = \frac{1001}{1000}, \ \frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting.

Quantitative Convergence for Sparse Ergodic Averages in $L^1$

TL;DR

This work addresses the challenge of ergodic convergence along sparse times at the endpoint. It develops a unified Calderón-Zygmund framework that treats both deterministic sparse sequences (with ) and random hitting-time sequences (with ), proving almost-sure convergence of for all . The authors introduce quantitative oscillation-variation measures (jump-counting, , and ) to obtain explicit convergence rates, extending Bourgain's technique and improving prior thresholds (e.g., up to in the deterministic case and the random setting). The results yield both universal -endpoint convergence and quantitative rate information, which enhances understanding of sparsity in ergodic averages and provides a versatile method for analyzing convergence across deterministic and probabilistic constructions with potential applications to sparse ergodic problems and related harmonic-analysis frameworks.

Abstract

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the endpoint. Specifically, suppose that where are Bernoulli random variables with expectations , and we restrict to . Then (almost surely) for any measure-preserving system, , and any , the ergodic averages converge -a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with , respectively, and LaVictoire, who established the random result, all in a non-quantitative setting.

Paper Structure

This paper contains 10 sections, 18 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Proof Strategy
  3. Preliminaries
  4. Notation
  5. Asymptotic Notation
  6. Calderón-Zygmund Theory
  7. Examples
  8. Deterministic Examples
  9. Random Examples
  10. Pointwise Convergence

Key Result

Theorem 1.4

Let $\mathbb{D} \subset \mathbb{N}$ be a $\lambda$-lacunary sequence of integers, thus $N'/N \geq \lambda > 1$ for all $N < N' \in \mathbb{D}$. Then for any $\epsilon > 0$, any $r > 2$, and any increasing sequence $\{ M_j \} \subset \mathbb{D}$, there exists an absolute constant $C_\lambda < \infty$ whenever $a_n = \lfloor n^c \rfloor, 1 < c < 8/7$, or whenever $a_n$ is as in e:hit with $0 < \alph

Theorems & Definitions (29)

  • Conjecture 1
  • Definition 1.2
  • Theorem 1.4
  • Theorem 1.7: Non-Quantitative Formulation
  • Proposition 2.5
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 19 more