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

Leonid Gorbunov

TL;DR

The paper extends Marstrand-type density results to a discrete, $m$-adic metric space by defining upper and lower $\alpha$-densities on the space of infinite sequences and introducing the density quotients $\mathcal{C}_{loc}(\mu,\alpha)$ and $\mathcal{C}(\mu,\alpha)$. It proves a universal lower bound for $\mathcal{C}_{loc}(\mu,\alpha)$ when $m^{\alpha}$ is not an integer and constructs a uniform measure achieving a sharp upper bound for $\mathcal{C}(\mu,\alpha)$, showing the bounds depend only on $m^{\alpha}$ and are dimension-free. The methods combine a discrete analogue of density analysis with a combinatorial marking-interval argument and a uniform-measure parametrization to obtain explicit, tight bounds. These results provide a discrete counterpart to the Marstrand–Preiss framework, offering precise density-control in a non-Euclidean setting suitable for further geometric-measure-theory investigations on hierarchical 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.

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

TL;DR

The paper extends Marstrand-type density results to a discrete, -adic metric space by defining upper and lower -densities on the space of infinite sequences and introducing the density quotients and . It proves a universal lower bound for when is not an integer and constructs a uniform measure achieving a sharp upper bound for , showing the bounds depend only on and are dimension-free. The methods combine a discrete analogue of density analysis with a combinatorial marking-interval argument and a uniform-measure parametrization to obtain explicit, tight bounds. These results provide a discrete counterpart to the Marstrand–Preiss framework, offering precise density-control in a non-Euclidean setting suitable for further geometric-measure-theory investigations on hierarchical 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.
Paper Structure (5 sections, 7 theorems, 19 equations)

This paper contains 5 sections, 7 theorems, 19 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Main results
  4. Proof of Theorem \ref{['mainlower']}
  5. Proof of Theorem \ref{['mainupper']}

Key Result

Theorem 1

Let $\mu$ be a Radon measure on $\mathop{\mathrm{\mathbb{R}}}\nolimits^d$, let $A$ be a set of positive measure, and let $\alpha \in [0, d]$ be a real number. Suppose that for $\mu$-a.e. $x \in A$ the $\alpha$-density of $\mu$ at $x$ exists, is finite and positive. Then $\alpha$ is integer.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1: Marstrand Marstrand, 1964
  • Theorem 2: Preiss, 1987
  • Definition 2
  • Remark
  • Remark
  • Theorem 3
  • Theorem 4
  • Definition 3
  • Lemma 1
  • ...and 7 more