  The aim of this paper is to show that the discrete maximal function
$$\mathcal{M}_{h}f(x)=\sup_{N\in\mathbb{N}}\frac{1}{|\mathbf{N}_{h}\cap[1,
N]|}\Big|\sum_{n\in \mathbf{N}_{h}\cap[1, N]}f(x-n)\Big|,\ \ \mbox{for
$x\in\mathbb{Z}$},$$ is of weak type $(1, 1)$, where
$\mathbf{N}_{h}=\{n\in\mathbb{N}: \exists_{m\in\mathbb{N}}\ n=\lfloor
h(m)\rfloor\}$ for an appropriate function $h$. As a consequence we also obtain
pointwise ergodic theorem along the set $\mathbf{N}_{h}$.
