Iterated Logarithmic Covering Growth and Grouped Multiresolution Spaces in Fractal Scale Systems

While classical fractal geometry studies the distribution of spatial structure across a uniform resolution parameter, fractal scale geometry additionally studies the distribution of the resolutions themselves and the laws coupling scale structure to spatial or functional complexity. Defining fractal scale systems as geometric sets, non-autonomous constructions, and multiscale function spaces governed by a prescribed sequence of logarithmic scales, we prove that three analytical operations obey three distinct sparsity thresholds. First, for non-autonomous Cantor sets with polynomially expanding scale gaps, two-sided covering bounds across all radii show that the box-counting dimension vanishes, the iterated logarithmic dimension equals the asymptotic scale-counting exponent, and the associated exponential-logarithmic Hausdorff gauge measure equals 2-1/2. Second, the lower- and upper-weighted grouped wavelet Sobolev spaces coincide with classical Sobolev spaces if and only if the scale gaps are bounded, and no single block-interpolated weight recovers equivalence when the gaps are unbounded. Third, unit asymptotic ratio of consecutive logarithmic scales is necessary and sufficient for scale-sampled box-counting dimensions to equal the classical dimension across all compact sets.