A generalization of Marstrand's theorem for projections of cartesian products
Annales de l'I.H.P. Analyse non linéaire, Tome 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 ,,K n be Borel subsets of m 1 ,, m n respectively, and π: m 1 ×× m n k be a surjective linear map. We set

𝔪:= min { iI dim H (K i )+ dim π iI c m i ,I{1,,n},I}.
Consider the space Λ m ={(t,O),t,O𝑆𝑂(m)} with the natural measure and set Λ=Λ m 1 ××Λ m n . For every λ=(t 1 ,O 1 ,,t n ,O n )Λ and every x=(x 1 ,,x n ) m 1 ×× m n we define π λ (x)=π(t 1 O 1 x 1 ,,t n O n x n ). Then we have Theorem (i) If 𝔪>k , then π λ (K 1 ××K n ) has positive k-dimensional Lebesgue measure for almost every λΛ . (ii) If 𝔪k and dim H (K 1 ××K n )= dim H (K 1 )++ dim H (K n ) , then dim H (π λ (K 1 ××K n ))=𝔪 for almost every λΛ .

DOI : 10.1016/j.anihpc.2014.04.002
Mots clés : 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},
     mrnumber = {3390086},
     zbl = {1321.28019},
     language = {en},
     url = {http://www.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
SP  - 833
EP  - 840
VL  - 32
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/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 http://www.numdam.org/articles/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, Tome 32 (2015) no. 4, pp. 833-840. doi : 10.1016/j.anihpc.2014.04.002. http://www.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

Cité par Sources :