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
@article{AIHPC_2015__32_4_833_0,
author = {L\'opez, Jorge Erick and Moreira, Carlos Gustavo},
title = {A generalization of {Marstrand's} theorem for projections of cartesian products},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {833--840},
publisher = {Elsevier},
volume = {32},
number = {4},
year = {2015},
doi = {10.1016/j.anihpc.2014.04.002},
zbl = {1321.28019},
mrnumber = {3390086},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.04.002/}
}
TY  - JOUR
AU  - López, Jorge Erick
AU  - Moreira, Carlos Gustavo
TI  - A generalization of Marstrand's theorem for projections of cartesian products
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
DA  - 2015///
SP  - 833
EP  - 840
VL  - 32
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.04.002/
UR  - https://zbmath.org/?q=an%3A1321.28019
UR  - https://www.ams.org/mathscinet-getitem?mr=3390086
UR  - https://doi.org/10.1016/j.anihpc.2014.04.002
DO  - 10.1016/j.anihpc.2014.04.002
LA  - en
ID  - AIHPC_2015__32_4_833_0
ER  - 
%0 Journal Article
%A López, Jorge Erick
%A Moreira, Carlos Gustavo
%T A generalization of Marstrand's theorem for projections of cartesian products
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 833-840
%V 32
%N 4
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2014.04.002
%R 10.1016/j.anihpc.2014.04.002
%G en
%F AIHPC_2015__32_4_833_0
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/

[1] M. Hochman, P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. Math. 175 no. 3 (2012), 1001 -1059 | MR | Zbl

[2] R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153 -155 | MR | Zbl

[3] J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. Lond. Math. Soc. (3) 4 (1954), 257 -302 | MR | Zbl

[4] P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn., Math. 1 (1975), 227 -244 | MR | Zbl

[5] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press (1995) | MR | Zbl

[6] C.G. Moreira, J.-C. Yoccoz, Stable intersection of regular cantor sets with large Hausdorff dimensions, Ann. Math. 154 no. 1 (2001), 45 -96 | MR | Zbl

[7] Y. Peres, W. Schalg, Smoothness of projections, Bernoulli convolutions, and the dimensions of exceptions, Duke Math. J. 102 no. 2 (2000), 193 -251 | MR | Zbl

[8] A. Schrijver, Theory of Linear and Integer Programming, Wiley–Interscience, Chichester (1986) | MR | Zbl

Cited by Sources: