A Discrete Marstrand Type Slicing Theorem with the Mass Dimension

Abstract

Discretized versions of some central questions in geometric measure theory have attracted recent attention; here 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 follows as a special case. We show that the natural slicing question in this setting is true with the mass dimension.