  Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product
symbolic sequence space $(\{0,\cdots,m_1-1\} \times
\{0,\cdots,m_2-1\})^{\mathbb{N}^*}$ that are invariant under the action of the
semigroup of multiplicative integers. These sets are defined following Kenyon,
Peres and Solomyak and using a fixed integer $q \geq 2$. We compute the
Hausdorff and Minkowski dimensions of the projection of these sets onto an
affine grid of the unit square. The proof of our Hausdorff dimension formula
proceeds via a variational principle over some class of Borel probability
measures on the studied sets. This extends well-known results on self-affine
Sierpinski carpets. However, the combinatoric arguments we use in our proofs
are more elaborate than in the self-similar case and involve a new parameter,
namely $j = \left\lfloor \log_q \left( \frac{\log(m_1)}{\log(m_2)} \right)
\right\rfloor$. We then generalize our results to the same subsets defined in
dimension $d \geq 2$. There, the situation is even more delicate and our
formulas involve a collection of $2d-3$ parameters.
