Weaker quantization dimension results for self-similar measures

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.

💡 Research Summary

The paper investigates the quantization dimension of self‑similar measures, focusing on situations where the underlying iterated function system (IFS) does not satisfy any classical separation condition (such as the strong separation condition (SSC) or the open set condition (OSC)), but a sub‑IFS obtained at a certain iteration level does satisfy a separation property. The authors develop a framework that allows one to deduce quantitative information about the full measure from the well‑behaved sub‑system.

First, the authors recall the standard definitions of the n‑th quantization error V_{n,r}(μ) of order r, the quantization dimension D_r(μ), and related notions such as quantization coefficients, lower and upper quantization dimensions, and the Lipschitz‑1 metric d_L on the space Ω(ℝ^m) of Borel probability measures. They also introduce the convolution of measures, translations, and the class L(ℝ^m) of finitely supported probability measures, which is dense in Ω(ℝ^m) with respect to d_L.

The core technical contribution begins with Theorem 3.1, which shows that if μ is the self‑similar measure associated with a weighted IFS (WIFS) I = {f_i, ρ_i} and μ_n is the self‑similar measure associated with a sub‑WIFS I_n obtained by fixing a finite word ξ and restricting to a subset Υ_n of the n‑letter words, then μ_n is absolutely continuous with respect to μ (μ_n ≪ μ). The proof uses the contractive nature of the associated Hutchinson operator P on the space of probability measures equipped with the Monge‑Kantorovich metric, together with the Banach fixed‑point theorem. Corollary 3.2 strengthens this result by proving that μ_n actually possesses an L^∞‑density with respect to μ.

From absolute continuity, Proposition 3.3 immediately yields the inequality D_r(μ_n) ≤ D_r(μ) and the analogous inequality for the lower Assouad dimension dim_L. This shows that any “good” sub‑system cannot have a larger quantization dimension than the original system.

The authors then present a concrete example (Remark 3.4) of an IFS that fails the OSC globally but whose three‑fold compositions {f_{11}, f_{21}, f_{31}} satisfy the SSC, illustrating the relevance of the sub‑IFS approach.

The main theorem concerning the exact value of the quantization dimension is Theorem 3.6. Assume the WIFS consists of similitudes with similarity ratios s_i and probabilities ρ_i. Suppose that for some finite word ξ, the sub‑IFS {f_{ηξ} : η ∈ Υ_n} satisfies the OSC for every n. Define κ_r as the unique solution of

  ∑_{i∈Λ} (ρ_i s_i^r)^{κ_r/(r+κ_r)} = 1.

Then the quantization dimension of the original self‑similar measure μ is

  D_r(μ) = min{κ_r, m}.

The proof adapts the Graf‑Luschgy method for measures satisfying the OSC to the present setting, using the monotonicity of the function x ↦ x^{r/(r+x)} and a contradiction argument based on the inequality κ_{n,r} ≤ D_r(μ). This result extends the classical formula to cases where only a sub‑IFS enjoys separation, showing that the global quantization dimension is still governed by the same pressure‑type equation.

The paper also explores the “order‑zero” case, i.e., quantization with respect to the geometric mean error. Lemma 3.9 proves that for any two finite measures μ and ν, D_0(μ+ν) = max{D_0(μ), D_0(ν)}. Lemma 3.10 shows that convolution with a fixed finitely supported measure does not change D_0, i.e., D_0(μ∗ν) = D_0(μ). Using these lemmas, Theorem 3.11 demonstrates that the sets

 Ω_0(ℝ^m) = {μ ∈ Ω(ℝ^m) : lim_{r→0+} D_r(μ) = D_0(μ) = dim_H(μ)}

and

 Ω_0^>(ℝ^m) = {μ ∈ Ω(ℝ^m) : lim_{r→0+} D_r(μ) = D_0(μ) > dim_H(μ)}

are both dense in Ω(ℝ^m). The construction relies on approximating an arbitrary measure by finitely supported measures (dense in d_L), then convolving with a measure from Ω_0 or Ω_0^> to adjust the order‑zero dimension while keeping the approximation arbitrarily close.

Finally, Lemma 3.12 and Remark 3.13 address convergence in total variation and the variation metric. If a sequence of measures {μ_n} is absolutely continuous with respect to a fixed λ and their densities converge in L¹(λ), then μ_n converges to a limit μ both in the d_L metric and in total variation. This provides a robust topological framework for the approximations used earlier.

In summary, the authors introduce the concept of “partial separation” to extend quantization dimension theory beyond the classical separation‑condition regime. They prove absolute continuity of sub‑measures, establish sharp upper bounds for D_r, and obtain an exact formula for D_r in terms of a pressure‑type equation even when only a sub‑IFS satisfies the OSC. Moreover, they develop a detailed approximation theory for the order‑zero quantization dimension, showing that both “regular” (D_0 = Hausdorff dimension) and “irregular” (D_0 > Hausdorff dimension) measures are dense in the space of all probability measures. These results broaden the applicability of quantization theory to fractal models with overlapping constructions and provide new tools for both theoretical investigations and practical applications such as signal compression and data quantization in high‑dimensional settings.