A Generalization of Marstrand’s Theorem and Some Geometric Applications

Abstract

Abstract In this paper we prove using quite elementary methods, with a combinatorial nature, two general results related to Marstrand’s projection theorem in a quite general formulation over metric spaces under a suitable transversality condition (the “projections" are in principle only continuous, and the transversality condition gives flexibility in exponents) - the result is flexible enough to, in particular, recover most of the classical Marstrand-like theorems. We also give some geometrical applications of our results - we give new (and simpler) proofs of a theorem by Solomyak on arithmetic sums of homogeneous Cantor sets and of result related to Falconer’s distance conjecture, and we also prove a version of Marstrand’s theorem for metrics of non-positive curvature in the plane.