Quantization Dimension and Stability for Infinite Self-Similar Measures with Respect to Geometric Mean Error

The paper proves that, for an infinite self-similar IFS under SSC with ∑ p_j log s_j convergent, the order-zero quantization dimension exists and equals the Hausdorff dimension (Theorem 3.1) via (i) approximation by finite subsystems plus continuity of the geometric-mean error to obtain the lower bound (Proposition 3.7), and (ii) a finite-system reduction using an auxiliary map covering the tail and a finite antichain estimate to obtain the upper bound (Proposition 3.8). The candidate solution follows the same two pillars. Minor gaps (e.g., handling the tail via a finite surrogate map T1 and avoiding a monotonicity claim for the truncated dimensions) can be filled by the paper’s lemmas and propositions, so the approaches are substantially the same and correct .