Potential method and projection theorems for macroscopic Hausdorff dimension

The macroscopic Hausdorff dimension DimH(E) of a set E⊂Rd was introduced by Barlow and Taylor to quantify “fractal at large scales” behavior unbounded, possibly discrete, sets E. We develop method based on potential theory in order estimate this Rd. Then, we apply obtain Marstrand-like projection theorems: given E⊂R2, for almost every θ∈[0,2π], the E straight line passing through 0 with angle θ has equal min⁡(DimH(E),1).