A generalization of Marstrand's theorem for projections of cartesian products
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 833-840.

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:Let ${K}_{1},\cdots ,{K}_{n}$ be Borel subsets of ${ℝ}^{{m}_{1}},\cdots ,{ℝ}^{{m}_{n}}$ respectively, and $\pi :{ℝ}^{{m}_{1}}×\cdots ×{ℝ}^{{m}_{n}}\to {ℝ}^{k}$ be a surjective linear map. We set

 $𝔪:=\mathrm{min}\left\{\sum _{i\in I}{\mathrm{dim}}_{H}\left({K}_{i}\right)+\mathrm{dim}\pi \left(\underset{i\in {I}^{c}}{⨁}{ℝ}^{{m}_{i}}\right),\phantom{\rule{0.166667em}{0ex}}I\subset \left\{1,\cdots ,n\right\},\phantom{\rule{0.166667em}{0ex}}I\ne ⌀\right\}.$
Consider the space ${\Lambda }_{m}=\left\{\left(t,O\right),\phantom{\rule{0.166667em}{0ex}}t\in ℝ,\phantom{\rule{0.166667em}{0ex}}O\in \mathrm{𝑆𝑂}\left(m\right)\right\}$ with the natural measure and set $\Lambda ={\Lambda }_{{m}_{1}}×\cdots ×{\Lambda }_{{m}_{n}}$. For every $\lambda =\left({t}_{1},{O}_{1},\cdots ,{t}_{n},{O}_{n}\right)\in \Lambda$ and every $x=\left({x}^{1},\cdots ,{x}^{n}\right)\in {ℝ}^{{m}_{1}}×\cdots ×{ℝ}^{{m}_{n}}$ we define ${\pi }_{\lambda }\left(x\right)=\pi \left({t}_{1}{O}_{1}{x}^{1},\cdots ,{t}_{n}{O}_{n}{x}^{n}\right)$. Then we have Theorem (i) If $𝔪>k$ , then ${\pi }_{\lambda }\left({K}_{1}×\cdots ×{K}_{n}\right)$ has positive k-dimensional Lebesgue measure for almost every $\lambda \in \Lambda$ . (ii) If $𝔪⩽k$ and ${\mathrm{dim}}_{H}\left({K}_{1}×\cdots ×{K}_{n}\right)={\mathrm{dim}}_{H}\left({K}_{1}\right)+\cdots +{\mathrm{dim}}_{H}\left({K}_{n}\right)$ , then ${\mathrm{dim}}_{H}\left({\pi }_{\lambda }\left({K}_{1}×\cdots ×{K}_{n}\right)\right)=𝔪$ for almost every $\lambda \in \Lambda$ .

DOI: 10.1016/j.anihpc.2014.04.002
Keywords: Fractal geometry, Hausdorff dimensions, Potential theory, Fourier transform, Dynamical systems
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title = {A generalization of {Marstrand's} theorem for projections of cartesian products},
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López, Jorge Erick; Moreira, Carlos Gustavo. A generalization of Marstrand's theorem for projections of cartesian products. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 833-840. doi : 10.1016/j.anihpc.2014.04.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.04.002/

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