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

18 pages; Clarified definitions in Section 2 which were not there in the earlier version

References in corpus (2)

• Marstrand-type theorems for the counting and mass dimensions in ${\mathbb{Z}}^d$
• Moreira's Theorem on the arithmetic sum of dynamically defined Cantor sets

Cited by in corpus (1)

• Marstrand type slicing statements in ${\mathbb{Z}}^2\subset {\mathbb{R}}^2$ are false for the counting dimension