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, p. 833-840
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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 : https://doi.org/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},
     publisher = {Elsevier},
     volume = {32},
     number = {4},
     year = {2015},
     pages = {833-840},
     doi = {10.1016/j.anihpc.2014.04.002},
     zbl = {1321.28019},
     mrnumber = {3390086},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIHPC_2015__32_4_833_0/

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