Exact asymptotic order for generalised adaptive approximations

TL;DR

The paper develops a unified abstract framework for adaptive approximations using a monotone set function $\mathfrak{J}$ on dyadic cubes and introduces $x$-good partitions $G_x$ with size $M(x)$. It defines the $\mathfrak{J}$-partition function $\tau_{\mathfrak{J}}(q)$ and critical exponents $\mathfrak{q}$ and $\kappa$, linking the upper and lower partition entropies to fractal-geometric quantities via sharp bounds and regularity conditions. Under MF-regular or PF-regular assumptions, the upper and lower entropies coincide with these exponents, yielding exact asymptotics for partition growth and dual quantities, with applications to Krein–Feller spectral problems, measure quantization, and Sobolev embedding widths. The work combines adaptive-partition algorithms, coarse multifractal analysis, and large-deviation techniques to provide precise, quantifiable rates and broad applicability across spectral theory and geometric measure contexts.

Abstract

In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function $\mathfrak{J}$ defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding $\mathfrak{J}$-partition function, and we are able to provide upper and lower bounds in term of fractal-geometric quantities. With properly chosen $\mathfrak{J}$, our new approach has applications in many different areas of mathematics, including the spectral theory of Krein-Feller operators, quantization dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gelfand and linear widths for Sobolev embeddings into $L_μ^p$-spaces.

Figures (3)

• Figure 1.1: Illustration of the adaptive approximation algorithm for $\mathfrak{J}(Q)\coloneqq\left(\nu\varotimes\nu\right)(Q)\left(\Lambda(Q)\right)^{2}$, $Q\in\mathcal{D}^{*}\coloneqq\left\{ Q\in\mathcal{D}:\mathfrak{J}\left(Q\right)>0\right\}$, $d=2$, where $\nu$ denotes the $\left(0.1,0.9\right)$-Cantor measure supported on the triadic Cantor set, i. e. the self similar measure generated by the IFS $S_{1}:x\mapsto x/3$, $S_{2}:x\mapsto x/3+2/3$ and probability weights $p_{1}=0.1$ and $p_{2}=0.9$ (see MR625600). Here, the light grey cubes belong to $G_{10^{-3}}$, the grey cubes belong to $G_{10^{-4}}$ and the black cubes belong to $G_{10^{-7}}$. Of course, the darker cubes overlay the lighter ones.
• Figure 1.2: A typical partition function $\tau$ with $\tau\left(0\right)=2$, $\tau\left(1\right)=1$ and $\dim_{\infty}\left(\mathfrak{J}\right)>0$. Natural bounds for $\overline{h}=\mathfrak{q}>1$ in this setting are the zeros of the dashed line $q\mapsto-q\left(\tau\left(0\right)-\tau\left(1\right)\right)+\tau\left(0\right)$ and the dotted line $q\mapsto\left(1-q\right)\dim_{\infty}\left(\mathfrak{J}\right)+\tau\left(1\right)$ as given in prop:Geometric_Bounsd.
• Figure 1.3: Illustration for the same example as in fig:PartitionAlgo of the cubes $\mathcal{N}_{\alpha}\left(n\right)$ (light grey) and $G_{2^{-\alpha n}}$ (black, or dark grey if covered by an element of $\mathcal{N}_{\alpha}\left(n\right)$) with $n=4$, $\alpha=5.734$.

Theorems & Definitions (55)

• Remark 1.1
• Lemma 1.2
• Lemma 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• proof
• Remark 1.7
• Theorem 1.8
• Proposition 1.9