A generalization of Marstrand's theorem and some geometric applications

TL;DR

This paper extends Marstrand's projection theorem to general metric spaces using elementary, combinatorial methods, and applies these results to derive new geometric insights, including progress on Falconer's distance conjecture.

Contribution

It introduces a flexible, general formulation of Marstrand's theorem applicable to metric spaces and demonstrates its utility through novel geometric applications.

Findings

Generalized Marstrand's theorem for metric spaces

New geometric applications including a result related to Falconer's distance conjecture

Elementary combinatorial methods used for proofs

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.