Distribution of $δ$-connected components of self-affine sponge of Lalley-Gatzouras type

Abstract: Let $(E, \rho)$ be a metric space and let $h_E\left( \delta \right)$ be the cardinality of the set of $\delta$-connected components of $E$. In literature, in case of that $E$ is a self-conformal set satisfying the open set condition or $E$ is a self-affine Sierpi\'nski sponge, necessary and sufficient condition is given for the validity of the relation $ h_E(\delta)\asymp \delta^{-\dim_B E}, \text{ when }\delta\to 0. $ In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure $\mu$ such that for any cylinder $R$, it holds that $ h_R(\delta)\asymp \mu(R) \delta^{-\dim_B E}, \text{ when }\delta\to 0. $