关于笛卡儿积投影的Marstrand定理的推广
A generalization of Marstrand's theorem for projections of cartesian products
摘要：
本文证明了Marstrand关于集合的笛卡儿积的投影定理的如下变形：设$K_1,...,K_n$分别是$\mathbb R^{m_1},... ,\mathbb R^{m_n}$的Borel子集，且$π:\mathbb R^{m_1}\times...\times\mathbb R^{m_n}\to\mathbb R^k$是满射线性映射.我们设置$$\mathfrak{m}:=\min\{\sum_{i\in I}\dim_H(K_i) + \dimπ(\bigoplus_{i\in I^c}\mathbb R^{m_i}), I\subset\{1,...,n\}, I\ne\emptyset\}.$$考虑空间$Λ_m=\{(t,O), t\in\mathbb R, O\in SO(m)\}$与自然测度和设置$Λ=Λ_{m_1}\times...\timesΛ_{m_n}$。对于每个$λ=(t_1,O_1,...,t_n,O_n)\inΛ$和每个$x=(x^1,,x^n)\in\mathbb R^{m_1}\times...\times\mathbb R^{m_n}$，我们定义$π_λ(x)=π(t_1O_1x^1,...,t_nO_nx^n)$。如果$\mathfrak{m}>k$，则$π_λ(K_1\times...\times K_n)$对几乎每个$λ\inΛ$都有正的$k$维Lebesgue测度.如果$\mathfrak{m}\leq k$和$\dim_H(K_1\times...\times K_n)=\dim_H(K_1)+...+\dim_H(K_n)$，则\dim_H（π_λ（K_1 \times. \times K_n））= \mathfrak{m} $ for almost every $ λ \in <$$。
英文摘要：
We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$ respectively, and $π:\mathbb R^{m_1}\times...\times\mathbb R^{m_n}\to\mathbb R^k$ be a surjective linear map. We set $$\mathfrak{m}:=\min\{\sum_{i\in I}\dim_H(K_i) + \dimπ(\bigoplus_{i\in I^c}\mathbb R^{m_i}), I\subset\{1,...,n\}, I\ne\emptyset\}.$$ Consider the space $Λ_m=\{(t,O), t\in\mathbb R, O\in SO(m)\}$ with the natural measure and set $Λ=Λ_{m_1}\times...\timesΛ_{m_n}$. For every $λ=(t_1,O_1,...,t_n,O_n)\inΛ$ and every $x=(x^1,,x^n)\in\mathbb R^{m_1}\times...\times\mathbb R^{m_n}$ we define $π_λ(x)=π(t_1O_1x^1,...,t_nO_nx^n)$. Then we have $(i)$ If $\mathfrak{m}>k$, then $π_λ(K_1\times...\times K_n)$ has positive $k$-dimensional Lebesgue measure for almost every $λ\inΛ$. $(ii)$ 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(π_λ(K_1\times...\times K_n))=\mathfrak{m}$ for almost every $λ\inΛ$.
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