Abstract
We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:Let K1,…,Kn be Borel subsets of Rm1,…,Rmn respectively, and π:Rm1×…×Rmn→Rk be a surjective linear map. We set m:=min{∑i∈IdimH(Ki)+dimπ(⨁i∈IcRmi),I⊂{1,…,n},I≠∅}. Consider the space Λm={(t,O),t∈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)∈Rm1×…×Rmn we define πλ(x)=π(t1O1x1,…,tnOnxn). Then we haveTheorem(i)Ifm>k, thenπλ(K1×…×Kn)has positive k-dimensional Lebesgue measure for almost everyλ∈Λ.(ii)Ifm⩽kanddimH(K1×…×Kn)=dimH(K1)+…+dimH(Kn), thendimH(πλ(K1×…×Kn))=mfor almost everyλ∈Λ.
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