Jorge Erick López | Searcharxiv

arXiv2021

A generalization of Marstrand's theorem

In this paper we prove two general results related to Marstrand's projection theorem in a quite general formulation over separable metric spaces under a suitable transversality hypothesis (the "projections" are in principle only measurable) - the result is flexible enough to, in particular, recover most of the classical Marstrand-like theorems. Our proofs use simpler tools than many classical works in the subject, where some techniques from harmonic analysis or special geometrical structures on the spaces are used.

math.MG↗

arXiv2011

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

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 $λ\inΛ$.

math.CA↗