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 $L_{\mu}^p$-spaces.

翻译：本文提出了一种抽象方法，用于研究关于dyadic立方体上定义的单调集函数$\mathfrak{J}$的自适应逼近的渐近阶数。我们通过相应$\mathfrak{J}$-配分函数的临界值确定了精确的上界，并能够用分形几何量给出上下界估计。在适当选取$\mathfrak{J}$的情形下，这一新方法可应用于数学的多个领域，包括Krein-Feller算子的谱理论、紧支概率测度的量化维数，以及Sobolev嵌入到$L_{\mu}^p$空间的Kolmogorov宽度、Gelfand宽度和线性宽度的精确渐近阶数。