THE DISTRIBUTION OF THE LARGEST DIGIT FOR PARABOLIC ITERATED FUNCTION SYSTEMS OF THE INTERVAL

The paper proves dim_H L(α) = dim_H Λ(Φ) for every α ∈ [0,∞] under (B1),(B2) by (i) constructing finite, uniformly contracting p-block IFSs Φ_p whose limit-set dimensions approximate dim_H Λ(Φ) from below, while still encoding parabolic behavior inside the blocks, and (ii) inserting large digits at carefully chosen times and pulling back via a Hölder map to transfer dimension (Proposition 3.2), completing the lower bound; the upper bound is immediate. Crucially, the paper emphasizes that one cannot simply exclude parabolic indices: the set of points whose digits are never parabolic has strictly smaller Hausdorff dimension than Λ(Φ) if a parabolic index exists, so parabolic indices cannot be ignored in establishing the equality (Theorem 1.1 and the accompanying discussion). The candidate solution’s core step assumes a finite sub-IFS consisting only of non-parabolic original indices can approximate dim_H Λ(Φ) arbitrarily well after removing parabolic indices; this directly contradicts the paper’s key observation and fails in general. The rest of the candidate’s construction (inserting sparse spikes and a Moran/Carathéodory lower bound) cannot rescue the argument once this approximation step is invalid. Hence the paper’s argument is correct, while the model’s is flawed on a decisive point. See Theorem 1.1 and the warning about not excluding parabolic indices, and the finite p-block construction (Proposition 2.4) used to approximate dimension from below .