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

López, Jorge Erick ; Moreira, Carlos Gustavo

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 833-840 / Harvested from Numdam

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 $ {\mathbb{R}}^{{m}_{1}},\dots ,{\mathbb{R}}^{{m}_{n}}$ respectively, and $ \pi :{\mathbb{R}}^{{m}_{1}}\times \dots \times {\mathbb{R}}^{{m}_{n}}\to {\mathbb{R}}^{k}$ be a surjective linear map. We set 𝔪:= min {∑ i∈I dim H (K i )+ dim π⨁ i∈I c ℝ m i ,I⊂{1,⋯,n},I≠⌀}. Consider the space $ {\Lambda }_{m}=\{(t,O),\,t\in \mathbb{R},\,O\in \mathit{SO}(m)\}$ with the natural measure and set $ \Lambda ={\Lambda }_{{m}_{1}}\times \dots \times {\Lambda }_{{m}_{n}}$. For every $ \lambda =({t}_{1},{O}_{1},\dots ,{t}_{n},{O}_{n})\in \Lambda $ and every $ x=({x}^{1},\dots ,{x}^{n})\in {\mathbb{R}}^{{m}_{1}}\times \dots \times {\mathbb{R}}^{{m}_{n}}$ we define $ {\pi }_{\lambda }(x)=\pi ({t}_{1}{O}_{1}{x}^{1},\dots ,{t}_{n}{O}_{n}{x}^{n})$. Then we have Theorem (i) If $ \mathfrak{m}>k$ , then $ {\pi }_{\lambda }({K}_{1}\times \dots \times {K}_{n})$ has positive k-dimensional Lebesgue measure for almost every $ \lambda \in \Lambda $ . (ii) If $ \mathfrak{m}⩽k$ and $ {\mathrm{dim}}_{H}({K}_{1}\times \dots \times {K}_{n})={\mathrm{dim}}_{H}({K}_{1})+\dots +{\mathrm{dim}}_{H}({K}_{n})$ , then $ {\mathrm{dim}}_{H}({\pi }_{\lambda }({K}_{1}\times \dots \times {K}_{n}))=\mathfrak{m}$ for almost every $ \lambda \in \Lambda $ .

Détail

Publié le : 2015-01-01

MR 3390086 | Zbl 1321.28019

DOI : https://doi.org/10.1016/j.anihpc.2014.04.002

Bibliographie

[1] M. Hochman, P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. Math. 175 no. 3 (2012), 1001 -1059 | MR 2912701 | Zbl 1251.28008

[2] R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153 -155 | MR 248779 | Zbl 0165.37404

[3] J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. Lond. Math. Soc. (3) 4 (1954), 257 -302 | MR 63439 | Zbl 0056.05504

[4] P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn., Math. 1 (1975), 227 -244 | MR 409774 | Zbl 0348.28019

[5] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press (1995) | MR 1333890 | Zbl 0819.28004

[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 1847588 | Zbl 1195.37015

[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 1749437 | Zbl 0961.42007

[8] A. Schrijver, Theory of Linear and Integer Programming, Wiley–Interscience, Chichester (1986) | MR 874114 | Zbl 0665.90063