Weak type $(1, 1)$ estimates for maximal functions along $1$-regular sequences of integers

TL;DR

This paper proves pointwise convergence of averages along 1-regular integer sequences, including sequences like \\lfloor n \\log n \\rfloor, extending classical results to more irregular sets.

Contribution

It establishes weak type (1,1) estimates for maximal functions along 1-regular sequences, a novel extension in harmonic analysis on integers.

Findings

Pointwise convergence of averages for \\ell^1(\\mathbb{Z}) functions.

Weak type (1,1) bounds for maximal functions along 1-regular sequences.

Includes sequences like \\lfloor n \\log n \\rfloor as examples.

Abstract

We show the pointwise convergence of the averages \[ \mathcal{A}_N f(x) = \frac{1}{\# \mathbf{B}_N} \sum_{n \in \mathbf{B}_N} f(x + n) \] for $f \in \ell^1(\mathbb{Z})$ where $\mathbf{B}_N = \mathbf{B} \cap [1, N]$, and $\mathbf{B}$ is a $1$-regular sequence of integers, for example $\mathbf{B} = \{\lfloor n \log n \rfloor : n \in \mathbb{N}\}$.