We consider subsets of the (symbolic) sequence space that are invariant under
the action of the semigroup of multiplicative integers. A representative
example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$
for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets
and show that they are typically different. The proof proceeds via a
variational principle for multiplicative subshifts.