Weaker quantization dimension results for self-similar measures
Saurabh Verma, Shivam Dubey
TL;DR
The paper studies the quantization dimension of self-similar measures under weak separation by analyzing a level‑$n$ sub‑IFS that satisfies separation. It proves that the sub‑IFS measure $\mu_n$ is absolutely continuous with respect to the original measure $\mu$, and that $D_r(\mu_n)\le D_r(\mu)$, with a fixed‑point contraction operator providing $\mu_n$ as the limit. In the OSC case for the sub‑IFS, the quantization dimension $D_r(\mu_n)$ is characterized by a unique $\kappa_{n,r}$ solving $\sum_{\eta \\in \\Upsilon^n} (\\varsigma_{\\eta} s_{\\xi \\eta}^r)^{\frac{\\\kappa_{n,r}}{r+\\ appa_{n,r}}}=1$, and $D_r$ is nondecreasing in $r$ with $D_0(\mu)$ recovered as $r\to 0^+$. The paper further develops an approximation framework showing that spaces of measures can be densely approached by subclasses with OSC, and it proves several stability properties: $D_0$ is subadditive under addition, unaffected by convolution, and preserved under limits, enabling dense approximation of $ ext{Ω}(\mathbb{R}^m)$. These results extend quantization-dimension analysis to weaker separation regimes and provide tools for approximating measure spaces with prescribed quantization characteristics.
Abstract
In this paper, we investigate the quantization dimension of self-similar measures, particularly when the IFS does not satisfy the separation condition, but the sub-IFS at some level satisfies the separation condition. Further, we study the approximation of the space of Borel probability measures $\mathcal{P}(\mathbb{R}^m)$ with respect to the geometric mean error, i.e., the quantization dimension of order zero.