A variant 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: 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*}