Quantization for a condensation system

TL;DR

This work analyzes quantization for condensation measures arising from condensation systems, focusing on two explicit cases with a two-map self-similarity and a base measure $\nu$ that is either discrete or uniform. By combining exact moment calculations, Voronoi-quantization analysis, and recursive constructions of optimal $n$-means, the authors establish the condensation dimension $D(\mu)$ as the maximum of the base dimension and a critical exponent $\kappa$, derive explicit optimal sets and quantization errors, and prove the nonexistence of the $D(\mu)$-dimensional quantization coefficient. Despite the nonexistence of the exact coefficient, both lower and upper quantization coefficients remain finite and positive, highlighting robust asymptotic regularity. In the discrete-nu case, $D(\mu)=\kappa=\dfrac{2\ln 2}{\ln 75 - \ln 2}\approx 0.3825$, while in the uniform-nu case, $D(\mu)=D(\nu)=1$, illustrating how the base measure governs the condensation-augmented dimension. The results advance the understanding of quantization for inhomogeneous self-similar (condensation) measures and provide explicit, verifiable constructions for optimal quantizers and their asymptotics under SSC/IOSC.

Abstract

For a given $r \in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, represents the rate at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exist, we define $\underline{D}_r(μ)$ and $\overline{D}_r(μ)$ as the lower and the upper quantization dimensions of $μ$ of order $r$, respectively. In this paper, we investigate the quantization dimension of the condensation measure $μ$ associated with a condensation system $(\{S_j\}_{j=1}^N, (p_j)_{j=0}^N, ν).$ We provide two examples: one where $ν$ is an infinite discrete distribution on $\mathbb{R}$, and one where $ν$ is a uniform distribution on $\mathbb{R}$. For both the discrete and uniform distributions $ν$, we determine the optimal sets of $n$-means, and calculate the quantization dimensions of condensation measures $μ$, and show that the $D_r(μ)$-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.

Theorems & Definitions (108)

• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Proposition 2.5
• Corollary 2.6
• Remark 2.7
• Lemma 2.8
• Lemma 2.9
• Lemma 2.10