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

We show the pointwise convergence of the averages $${\mathcal{A}}_{N}f(x)=\frac{1}{\mathrm{\#}{\mathbf{B}}_N}\sum _{n\in {\mathbf{B}}_N}f(x+n)$$ for $f \in \ell^1($\mathbb{Z}$)$where$$\mathbf{B}$_N = $\mathbf{B}$ \cap [1, N]$,and$$\mathbf{B}$$isa$1$-regularsequenceofintegers,forexample$$\mathbf{B}$ = {\lfloor n \log n \rfloor : n \in $\mathbb{N}$}$.