A variant of Marstrand's theorem for projections of cartesian products

TL;DR

This paper extends Marstrand's projection theorem to Cartesian products of sets, establishing conditions under which projections have positive measure or specific Hausdorff dimension, based on the sets' dimensions and a parameter involving projections.

Contribution

It introduces a novel variant of Marstrand's theorem for Cartesian products, providing new criteria for measure and dimension of projections in higher-dimensional spaces.

Findings

Proves that projections have positive measure when a certain dimension sum exceeds a threshold.

Establishes the Hausdorff dimension of projections equals a computed minimum under specific conditions.

Extends classical projection results to more complex product sets with geometric measure considerations.

Abstract

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Consider the space $\Lambda_m=\set{(t,O), t\in\R, O\in SO(m)}$ with the natural measure and set $\Lambda=\Lambda_{m_1}\times\ppp\times\Lambda_{m_n}$. For every $\la=(t_1,O_1,\ppp,t_n,O_n)\in\Lambda$ and every $x=(x^1,\ppp,x^n)\in\R^{m_1}\times\ppp\times\R^{m_n}$ we define $\pi_\la(x)=\pi(t_1O_1x^1,\ppp,t_nO_nx^n)$. Suppose that $\pi$ is surjective and set $$\mathfrak{m}:=\min\set{\sum_{i\in I}\dim_H(K_i) + \dim\pi(\bigoplus_{i\in I^c}\R^{m_i}), I\subset\set{1,\ppp,n}, I\ne\emptyset}.$$ Then we have {thm*} \emph{(i)} If $\mathfrak{m}>k$, then $\pi_\la(K_1\times\ppp\times K_n)$ has positive $k$-dimensional Lebesgue measure for almost every $\la\in\Lambda$. \emph{(ii)} If $\mathfrak{m}\leq k$ and $\dim_H(K_1\times\ppp\times K_n)=\dim_H(K_1)+\ppp+\dim_H(K_n)$, then $\dim_H(\pi_\la(K_1\times\ppp\times K_n))=\mathfrak{m}$ for almost every $\la\in\Lambda$. {thm*}