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In this note, we present an abstract approach to study asymptotic orders foradaptive approximations with respect to a monotone set function $\mathfrak{J}$defined on dyadic cubes. We determine the exact upper order in terms of thecritical value of the corresponding $\mathfrak{J}$-partition function, and weare able to provide upper and lower bounds in term of fractal-geometricquantities. With properly chosen $\mathfrak{J}$, our new approach hasapplications in many different areas of mathematics, including the spectraltheory of Krein-Feller operators, quantization dimensions of compactlysupported probability measures, and the exact asymptotic order for Kolmogorov,Gelfand and linear widths for Sobolev embeddings into $L_{\mu}^p$-spaces.

Exact asymptotic order for generalised adaptive approximations