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

TL;DR

This paper extends Marstrand's theorem to projections of Cartesian products of sets in Euclidean spaces, establishing conditions under which the projections have positive measure or specific Hausdorff dimension.

Contribution

It introduces a generalized projection theorem for Cartesian products, providing new measure and dimension results for almost all rotations and scalings.

Findings

Projections have positive Lebesgue measure when a certain dimension sum exceeds target dimension.

Hausdorff dimension of projections equals a computed minimum under specified conditions.

Results hold for almost every combination of rotations and scalings in the parameter space.

Abstract

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