Quantization dimensions of compactly supported probability measures via Rényi dimensions

TL;DR

This work establishes a precise link between the upper quantization dimension $\bar{D}_{r}(\nu)$ of a compactly supported measure and the $L^{q}$-spectrum via the intersection point $q_{r}$ with the line of slope $r$, proving $\bar{D}_{r}(\nu)=\overline{\mathfrak{R}}_{\nu}(q_{r})=\frac{rq_{r}}{1-q_{r}}$ and showing continuity of $r \mapsto \bar{D}_{r}(\nu)$. It develops a partition-function framework and coarse multifractal formalism to derive tight upper and lower bounds for the quantization dimension, and demonstrates existence results for the quantization dimension for Gibbs measures with respect to conformal IFS without separation conditions as well as for inhomogeneous self-similar measures under iOSC. The analysis unifies quantization with Rényi-type spectra, providing practical criteria via $L^{q}$-regularity and partition entropies to determine $\bar{D}_{r}$ and its continuity, and yields refinements of existing fractal-dimension bounds. The results have potential implications for numerical quantization schemes and deepen the connections between quantization and fractal geometry, including self-conformal and overlapped systems.

Abstract

We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension $\bar{D}_{r}(ν)$ of order $r>0$ for an arbitrary compactly supported Borel probability measure $ν$ is given by its Rényi dimension at the point $q_{r}$ where the $L^{q}$-spectrum of $ν$ and the line through the origin with slope $r$ intersect. In particular, this proves the continuity of $r\mapsto\bar{D}_{r}(ν)$ as conjectured by Lindsay (2001). This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the $L^{q}$-spectrum restricted to $\left(0,1\right)$. We give sufficient conditions in terms of the $L^{q}$-spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a $\mathcal{C}^{1}$-self- conformal iterated function system on $\mathbb{R}^{d}$ without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.

Figures (1)

• Figure 1.1: The $L^{q}$-spectrum $\beta_{\nu}$ for the self-similar measure $\nu$ supported on the Sierpiński tetraeder in $\mathbb{R}^{3}$ with all four contraction ratios equal $1/2$ and with probability vector $\left(0.66,0.2,0.08,0.06\right)$. $\beta_{\nu}\left(0\right)=\overline{\dim}_{M}\left(\nu\right)=2$. For $r=2.3$ (slope of the dashed line) the intersection of the graph of $\beta_{\nu}$ and the dashed line determines $q_{r}$. The dotted line through $\left(q_{r},\beta_{\nu}\left(q_{r}\right)\right)$ and $\left(0,1\right)$ intersects the vertical axis in $\overline{D}_{r}\left(\nu\right)$.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Corollary 1.5: Continuity
• Corollary 1.6: Finite stability
• Proposition 1.7
• Remark 1.8
• Corollary 1.9
• Definition 1.10