VECTOR-VALUED FRACTAL FUNCTIONS: FRACTAL DIMENSION AND FRACTIONAL CALCULUS

The paper’s Theorem 3.7 establishes r ≤ dim_H(G(h)) ≤ R for the graph of a vector-valued FIF under two-sided Lipschitz bounds, proving the upper bound by a standard covering argument and the lower bound by constructing IFSs that satisfy SSC via a careful SOSC-based reduction; the contradiction step is sound and closes the argument (see Theorem 3.7 and its proof ; details around SOSC/SSC and the construction appear across the same section , with background on OSC/SOSC/SSC in Section 2.1 ). By contrast, the candidate solution’s lower-bound argument asserts µ(W_σ K) = c_σ^r and appeals to a “standard” Frostman estimate without invoking any separation or overlap control; this is not generally valid for bi-Lipschitz IFSs with overlaps, and the claim that no separation hypothesis is needed is false in this generality. The paper correctly supplies separation via SOSC→SSC (on a derived IFS) to justify the lower bound; the model omits this and therefore its proof is flawed, even though its final inequality matches the paper’s statement.