2011/07/02 by Jorge Erick López, López, Jorge Erick, Carlos Gustavo Moreira +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #math.CA #math.DS
paper · pdf · doi:10.48550/arxiv.1107.0424
8 pages
openalex publication_date 2011/07/02 · arxiv created 2011/07/03 · arxiv updated 2011/07/05 · openalex created_date 2022/09/29 · openalex updated_date 2026/07/28
We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Let K1,...,Kn Borel subsets of \mathbb Rm1,... ,\mathbb Rmn respectively, and π:\mathbb Rm1×...×\mathbb Rmn→\mathbb Rk be a surjective linear map. We set \mathfrakm:=min\∑i∈ IdimH(Ki) + dimπ(\bigoplusi∈ Ic\mathbb Rmi), I⊂\1,...,n\, I≠∅\. Consider the space Λm=\(t,O), t∈\mathbb R, O∈ SO(m)\ with the natural measure and set Λ=Λm1×...×Λmn. For every λ=(t1,O1,...,tn,On)∈Λ and every x=(x1,,xn)∈\mathbb Rm1×...×\mathbb Rmn we define πλ(x)=π(t1O1x1,...,tnOnxn). Then we have (i) If \mathfrakm>k, then πλ(K1×...× Kn) has positive k-dimensional Lebesgue measure for almost every λ∈Λ. (ii) If \mathfrakm≤ k and dimH(K1×...× Kn)=dimH(K1)+...+dimH(Kn), then dimH(πλ(K1×...× Kn))=\mathfrakm for almost every λ∈Λ.