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Asymptotic order of the quantization error for a class of self-similar measures with overlaps

Sanguo Zhu

TL;DR

The paper analyzes the asymptotics of the quantization error for self-similar measures generated by equi-contractive IFS with overlaps under a finite-type condition and total self-similarity. By leveraging Feng’s net-interval partitioning and a matrix-analytic framework, it proves that the upper and lower quantization coefficients at the quantization dimension $D_r(\mu)$ are strictly positive and finite, extending Graf and Luschgy’s OSC-based results to overlapping settings. The approach identifies a unique pressure-zero $s_r$ giving $D_r(\mu)=s_r$ and connects the global measure to a localized conditional measure via self-similarity to transfer estimates. The results apply to a broad class of measures, including Erdős measure, the 3-fold Cantor convolution, and certain $\lambda$-Cantor measures, highlighting the practical reach to overlapping fractal structures. Overall, the work advances the understanding of exact quantization-order in overlaps and provides verifiable conditions under which the quantization coefficients are well-behaved.

Abstract

Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $μ$ associated with $\{f_i\}_{i=1}^N$ and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for $μ$ are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures with the open set condition. Our result is applicable to a significant class of self-similar measures with overlaps, including Erdös measure, the $3$-fold convolution of the classical Cantor measure and the self-similar measures on some $λ$-Cantor sets.

Asymptotic order of the quantization error for a class of self-similar measures with overlaps

TL;DR

The paper analyzes the asymptotics of the quantization error for self-similar measures generated by equi-contractive IFS with overlaps under a finite-type condition and total self-similarity. By leveraging Feng’s net-interval partitioning and a matrix-analytic framework, it proves that the upper and lower quantization coefficients at the quantization dimension are strictly positive and finite, extending Graf and Luschgy’s OSC-based results to overlapping settings. The approach identifies a unique pressure-zero giving and connects the global measure to a localized conditional measure via self-similarity to transfer estimates. The results apply to a broad class of measures, including Erdős measure, the 3-fold Cantor convolution, and certain -Cantor measures, highlighting the practical reach to overlapping fractal structures. Overall, the work advances the understanding of exact quantization-order in overlaps and provides verifiable conditions under which the quantization coefficients are well-behaved.

Abstract

Let be a set of equi-contractive similitudes on satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures associated with and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures with the open set condition. Our result is applicable to a significant class of self-similar measures with overlaps, including Erdös measure, the -fold convolution of the classical Cantor measure and the self-similar measures on some -Cantor sets.
Paper Structure (12 sections, 6 theorems, 42 equations)

This paper contains 12 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Asymptotics of the quantization errors
  3. Equi-contractive IFS and finite-type condition
  4. Statement of the main result
  5. Preliminaries
  6. Net intervals and characteristic vectors
  7. Admissible words
  8. Measures of net intervals
  9. Some of Feng's results
  10. Some remarks
  11. Proof of Theorem \ref{['FTC']}
  12. Examples

Key Result

Theorem 1.1

Let $(f_i)_{i=1}^N$ be as defined in (equiifs) satisfying the FTC. Let $E$ denote the self-similar set determined by $(f_i)_{i=1}^N$ and $\mu$ the self-similar measure associated with $(f_i)_{i=1}^N$ and a positive probability vector $(p_i)_{i=1}^N$. Assume that $E$ is totally self-similar. Then for

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 8 more