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We prove a Marstrand type slicing theorem for the subsets of the integersquare lattice. This problem is the dual of the corresponding projectiontheorem, which was considered by Glasscock, and Lima and Moreira, with the massand 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, andthe result for subsets of the integer lattice follow as a special case. We showthat the natural slicing question in this setting is true with the massdimension.

A Marstrand type slicing theorem for subsets of $\mathbb{Z}^2 \subset \mathbb{R}^2$ with the mass dimension