Dimensions of "self-affine sponges" invariant under the action of multiplicative integers

Abstract

Let $m_1\ge m_2\ge 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\cdots ,m_1-1\}\times \{0,\cdots ,m_2-1\}{)}^{{\mathbb{N}}^{\ast }}$ 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 $q\ge 2$. 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 $j=\lfloor {\log }_q\left(\frac{\log (m_1)}{\log (m_2)}\right)\rfloor$. We then generalize our results to the same subsets defined in dimension $d\ge 2$. There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.