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

TL;DR

This paper establishes a Marstrand type slicing theorem for 1-separated subsets of the plane, including the integer lattice, demonstrating that the natural slicing property holds with respect to the mass dimension.

Contribution

It extends the Marstrand slicing theorem to 1-separated subsets of the plane, generalizing previous results on the dual projection theorem for lattice subsets.

Findings

Proves slicing theorem for 1-separated subsets of the plane.

Shows the theorem applies to subsets of the integer lattice.

Demonstrates the result with respect to the mass dimension.

Abstract

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.