Quantization dimension of probability measures supported on Cantor-like sets

Abstract

Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied.

Optimal Quantization for Uniform Distributions on Cantor-Like Sets

“…[15], Theorem 1.6 (3), respectively [17], Proposition 5.1) abstained from (35). Zhu [24] also characterized the quantization dimension for a large class of singular distributions on higher dimensional Cantor-like sets. Due to this more general approach, he needed the boundary condition (35) there.…”

“…The techniques developed by Zhu [24] need the boundary condition inf i∈N c i > 0 and therefore cannot be used here. Moreover, it remains unanswered whether it is possible to find conditions, which are equivalent to (36).…”

“…The construction of E m (V ) and relation (24) implies E m (V ) ⊂ V . Using (39) and Lemma 3, we obtain…”

“…[19], Example 5.5). Later on, the existence of the quantization dimension for distributions on (not necessarily self-similar) Cantor-like sets was systematically studied and characterized by Kesseböhmer and Zhu [15], Kreitmeier [17] and Zhu [24].…”

“…Proposition 5). Although the formula for the quantization dimension is a special case of the results of Zhu [24], the class of distributions considered in this present paper is not completely embraced by the one studied in [24] (cf. section 3.2).…”