Asymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets. (English) Zbl 1434.28010
Summary: Let \(E\) be a Bedford-McMullen carpet associated with a set of affine mappings \(\{ f_{i j} \}_{( i , j ) \in G}\) and let \(\mu\) be the self-affine measure associated with \(\{ f_{i j} \}_{( i , j ) \in G}\) and a probability vector \(( p_{i j} )_{( i , j ) \in G}\). 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 \(\{ f_{i j} \}_{( i , j ) \in G}\), we prove that the \(k\) th geometric error for \(\mu\) is of the same order as \(k^{- 1 / s_0}\).
MSC:
| 28A78 | Hausdorff and packing measures |
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