Potential method and projection theorems for macroscopic Hausdorff dimension

Abstract

The macroscopic Hausdorff dimension DimH(E) of a set E⊂Rd 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 Rd. Then, we apply this method to obtain Marstrand-like projection theorems: given a set E⊂R2, for almost every θ∈[0,2π], the projection of E on the straight line passing through 0 with angle θ has dimension equal to min⁡(DimH(E),1).