  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.
