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 R n such that all whose projections onto l-dimensional planes are exactly k-dimensional? We construct in R n 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.