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

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 $$\mathfrak{m}:=\min \{\sum _{i\in I}{\dim }_H(K_i)+\dim \pi (\underset{i\in I^c}{⨁}{\mathbb{R}}^{m_i}),I\subset \{1,...,n\},I\ne \varnothing \}.$$ 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 ${\pi }_{\lambda }(x)=\pi (t_1{O}_1{x}^1,...,t_n{O}_n{x}^n)$. 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 $\mathfrak{m}\le k$ and ${\dim }_H(K_1\times ...\times K_n)={\dim }_H(K_1)+...+{\dim }_H(K_n)$, then \dim_H(π_λ(K_1\times...\times K_n))=\mathfrak{m}$foralmostevery$λ\inΛ$.

8 pages