Dimensional Analysis of Fractal Interpolation Functions

The paper’s Theorem 3.14 states exactly the claim dim_B Graph(f_α) = 2 − s under the Hölder/oscillation hypotheses and the smallness condition ‖α‖_H < a^s min{1, (K_f − (‖b‖∞+M)k_α a^{−s})/(k_{f_α}+k_b)}; its proof establishes an upper bound via Hölder regularity (Theorem 3.12) and a lower bound via a quantitative oscillation inequality derived from the self-referential equation, then a covering count to get N(δ) ≳ δ^{-(2−s)} (equations around (3.2)–(3.3) and the subsequent K > 0 estimate) . The candidate solution reproduces the same structure: (1) shows f_α ∈ H^s with an a priori bound; (2) gives the standard s-Hölder upper bound dim_B ≤ 2−s; (3) derives a positive lower oscillation constant K′ with the same dependence on k_{f_α}, k_b, k_α, M; and (4) performs a lower-bound covering argument yielding dim_B ≥ 2−s. The only (minor) difference is that the model’s Step 4 uses disjoint interior windows instead of the paper’s δ-grid sum; both yield the same asymptotics. Thus both are correct and essentially the same proof idea. Minor presentational gaps in the paper’s lower-bound counting (handling intervals that straddle partition boundaries) can be fixed by excluding O(1) boundary strips without affecting the dimension conclusion .