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

Potential method and projection theorems for macroscopic Hausdorff dimension