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 $K_1,\dots ,K_n$ be Borel subsets of ${\mathbb{R}}^{m_1},\dots ,{\mathbb{R}}^{m_n}$ respectively, and $\pi :{\mathbb{R}}^{m_1}\times \cdots \times {\mathbb{R}}^{m_n}\to {\mathbb{R}}^k$ be a surjective linear map. We set

Consider the space ${\Lambda }_m=\{(t,O),t\in \mathbb{R},O\in \mathrm{SO}(m)\}$ with the natural measure and set $\Lambda ={\Lambda }_{m_1}\times \cdots \times {\Lambda }_{m_n}$. For every $\lambda =(t_1,O_1,\dots ,t_n,O_n)\in \Lambda$ and every $x=(x^1,\dots ,x^n)\in {\mathbb{R}}^{m_1}\times \cdots \times {\mathbb{R}}^{m_n}$ we define ${\pi }_{\lambda }(x)=\pi (t_1{O}_1{x}^1,\dots ,t_n{O}_n{x}^n)$. Then we have

Theorem.

(i) If $\mathfrak{m}>k$, then ${\pi }_{\lambda }(K_1\times \cdots \times K_n)$ has positive k-dimensional Lebesgue measure for almost every $\lambda \in \Lambda$.

(ii) If $\mathfrak{m}⩽k$ and ${\dim }_H(K_1\times \cdots \times K_n)={\dim }_H(K_1)+\cdots +{\dim }_H(K_n)$, then ${\dim }_H({\pi }_{\lambda }(K_1\times \dots \times K_n))=\mathfrak{m}$ for almost every $\lambda \in \Lambda$.