A generalization of Marstrand's theorem and some geometric applications

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 new geometrical applications of our results - one of them is a new result related to Falconer's distance conjecture.

Figures (3)

• Figure 1: $\gamma_{xy}$ does not pass through $p$
• Figure 2: $I\in [u,v]$.
• Figure 3: $I\notin [u,v]$.

Theorems & Definitions (35)

• Theorem 1.1
• Lemma 1.2: Howroyd
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm_Marstrand2']}
• Corollary 2.3
• proof : Proof of Theorem \ref{['thm_L2']}