Asymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets

Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings ${f_{ij}}_{(i,j)in G}$ and let $mu$ be the self-affine measure associated with ${f_{ij}}_{(i,j)in G}$ and a probability vector $(p_{ij})_{(i,j)in G}$. We study the asymptotics of the geometric mean error in the quantization for $mu$. Let $s_0$ be the Hausdorff dimension for $mu$. Assuming a separation condition for ${f_{ij}}_{(i,j)in G}$, we prove that the $n$th geometric error for $mu$ is of the same order as $n^{-1/s_0}$.