As demonstrated by the unique electronic properties of nanostructured materials, which are qualitatively different from the bulk properties of the same materials, there should be a general relationship between the dimensions of the sample structures and the physical/chemical properties. However, it is demanding to examine this intriguing problem experimentally, because one cannot prepare a series of samples based on the same material with systematically varying and noninteger dimensions. This problem is solved by considering the fractal dimensions of samples. The electronic structures of a series of powder samples of the high-TC superconductor YBa2Cu3O7-δ with the fractal dimensions (Df) of 2.5-3.0 were investigated using the electron spin resonance (ESR) spectra at 293 K, to determine a general and quantitative relationship between the electronic properties and fractal dimensions of the samples. The observed Df-dependences of ESR parameters, such as g-values and linewidths, were quantitatively consistent with those of the critical temperatures, critical current densities, and lower critical magnetic fields, all of which exhibited anomalies at Df ∼ 2.9. Considering the geometrical features of the fractal models, a hypothesis for explaining the observed Df-dependences of the cooperative phenomena has been proposed, suggesting a universal structural instability at particular fractal dimensions, which affects all the physical and chemical properties of the samples.