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

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:Let K1,…,KnK1,…,Kn be Borel subsets of Rm1,…,RmnRm1,…,Rmn respectively, and π:Rm1×…×Rmn→Rkπ:Rm1×…×Rmn→Rk be a surjective linear map. We setm:=min⁡{∑i∈IdimH⁡(Ki)+dim⁡π(⨁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×…×ΛmnΛ=Λm1×…×Λmn. For every λ=(t1,O1,…,tn,On)∈Λλ=(t1,O1,…,tn,On)∈Λ and every x=(x1,…,xn)∈Rm1×…×Rmnx=(x1,…,xn)∈Rm1×…×Rmn we define πλ(x)=π(t1O1x1,…,tnOnxn)πλ(x)=π(t1O1x1,…,tnOnxn). Then we haveTheorem. (i)If  m>km>k, then  πλ(K1×…×Kn)πλ(K1×…×Kn)has positive k-dimensional Lebesgue measure for almost every  λ∈Λλ∈Λ.(ii)If  m⩽km⩽kand  dimH⁡(K1×…×Kn)=dimH⁡(K1)+…+dimH⁡(Kn)dimH⁡(K1×…×Kn)=dimH⁡(K1)+…+dimH⁡(Kn), then  dimH⁡(πλ(K1×…×Kn))=mdimH⁡(πλ(K1×…×Kn))=mfor almost every  λ∈Λλ∈Λ.