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

López, Jorge Erick; Moreira, Carlos Gustavo

doi:10.1016/j.anihpc.2014.04.002

López, Jorge Erick ; Moreira, Carlos Gustavo

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 833-840

Résumé

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:Let $K_{1}, \dots, K_{n}$ be Borel subsets of $ℝ^{m_{1}}, \dots, ℝ^{m_{n}}$ respectively, and $π : ℝ^{m_{1}} \times \dots \times ℝ^{m_{n}} \to ℝ^{k}$ be a surjective linear map. We set

𝔪 : = \min {\sum_{i \in I} \dim_{H} (K_{i}) + \dim π (⨁_{i \in I^{c}} ℝ^{m_{i}}), I \subset {1, \dots, n}, I \neq ⌀} .

Consider the space

Λ_{m} = {(t, O), t \in ℝ, O \in 𝑆𝑂 (m)}

with the natural measure and set

Λ = Λ_{m_{1}} \times \dots \times Λ_{m_{n}}

. For every

λ = (t_{1}, O_{1}, \dots, t_{n}, O_{n}) \in Λ

and every

x = (x^{1}, \dots, x^{n}) \in ℝ^{m_{1}} \times \dots \times ℝ^{m_{n}}

we define

π_{λ} (x) = π (t_{1} O_{1} x^{1}, \dots, t_{n} O_{n} x^{n})

. Then we have Theorem (i) If

𝔪 > k

, then

π_{λ} (K_{1} \times \dots \times K_{n})

has positive k-dimensional Lebesgue measure for almost every

λ \in Λ

. (ii) If

𝔪 ⩽ k

and

\dim_{H} (K_{1} \times \dots \times K_{n}) = \dim_{H} (K_{1}) + \dots + \dim_{H} (K_{n})

, then

\dim_{H} (π_{λ} (K_{1} \times \dots \times K_{n})) = 𝔪

for almost every

λ \in Λ

.

MR Zbl

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},
     year = {2015},
     publisher = {Elsevier},
     volume = {32},
     number = {4},
     doi = {10.1016/j.anihpc.2014.04.002},
     mrnumber = {3390086},
     zbl = {1321.28019},
     language = {en},
     url = {https://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  - https://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 https://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

Bibliographie
Cité par

[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 :