A version of Marstrand's theorem on a discrete metric space

TL;DR

This paper adapts Marstrand's theorem to discrete metric spaces, providing explicit density estimates for measures, which advances understanding of geometric measure theory in discrete settings.

Contribution

It introduces a discrete version of Marstrand's theorem along with explicit density estimates, extending classical results to discrete metric spaces.

Findings

Established a discrete analogue of Marstrand's theorem.

Derived explicit upper and lower density estimates.

Enhanced understanding of measure behavior in discrete metric spaces.

Abstract

We present and prove the version of Marstrand's theorem for discrete metric space. We provide explicit estimates of the quotient of upper and lower densities of measures on this space.