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