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

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 $\mathbf{B} = {\lfloor n \log n \rfloor : n \in \mathbb{N}}$.