Potential method and projection theorems for macroscopic Hausdorff dimension

TL;DR

This paper introduces a potential theory-based method to estimate the macroscopic Hausdorff dimension of sets in R^d and applies it to prove projection theorems similar to Marstrand's, showing that most projections preserve dimension.

Contribution

Develops a new potential theory approach to estimate macroscopic Hausdorff dimension and establishes Marstrand-like projection theorems for unbounded sets in R^2.

Findings

Method effectively estimates macroscopic Hausdorff dimension.

Almost all projections preserve the dimension as min(Dim H(E), 1).

Provides a new tool for analyzing large-scale fractal properties.

Abstract

The macroscopic Hausdorff dimension Dim H (E) of a set E $\subset$ R d was introduced by Barlow and Taylor to quantify a "fractal at large scales" behavior of unbounded, possibly discrete, sets E. We develop a method based on potential theory in order to estimate this dimension in R d. Then, we apply this method to obtain Marstrand-like projection theorems: given a set E $\subset$ R 2 , for almost every $\theta$ $\in$ [0, 2$\pi$], the projection of E on the straight line passing through 0 with angle $\theta$ has dimension equal to min(Dim H (E) , 1).