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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 $μ$ 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 $μ$. Let $s_0$ be the Hausdorff dimension for $μ$. Assuming a separation condition for $\{f_{ij}\}_{(i,j)\in G}$, we prove that the $n$th geometric error for $μ$ is of the same order as $n^{-1/s_0}$.