  We prove a Marstrand type slicing theorem for the subsets of the integer
square lattice. This problem is the dual of the corresponding projection
theorem, which was considered by Glasscock, and Lima and Moreira, with the mass
and counting dimensions applied to subsets of $\mathbb{Z}^{d}$. In this paper,
more generally we deal with a subset of the plane that is $1$ separated, and
the result for subsets of the integer lattice follow as a special case. We show
that the natural slicing question in this setting is true with the mass
dimension.
