On Cantor sets with shadows of prescribed dimension

We consider a question raised by John Cobb: given positive integers n>l>k is there a Cantor set in Rn such that all whose projections onto l-dimensional planes are exactly k-dimensional? We construct in Rn a Cantor set such that all its shadows (projections onto hyperplanes) are k-dimensional for every 0⩽k⩽n−1. We also consider the extension of Cobbʼs question to Hilbert space.