Weak type (1, 1) inequalities for discrete rough maximal functions

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|,\ \ for $x\in\mathbb{Z$}, is of weak type $(1,1)$, where $$\mathbf{N}${h}={n\in$\mathbb{N}$: \exists{m\in$\mathbb{N}$}\ n=\lfloor h(m)\rfloor}$foranappropriatefunction$h$. As a consequence we also obtain pointwise ergodic theorem along the set ${\mathbf{N}}_h$.