On the weakness of the vague specification property

The paper proves that a topological dynamical system has the vague specification property (VSP) if and only if it has the asymptotic average shadowing property (AASP) (Theorem 4.6), by showing first that asymptotic average pseudo-orbits coincide with vague pseudo-orbits (Theorem 3.13) and then using the equivalence between density-1 tracing and Besicovitch-average tracing (Lemmas 2.2–2.3) to pass between VSP and AASP. The candidate solution reproduces the same two-way implication with essentially the same ingredients: (i) a density-of-good-blocks lemma, (ii) uniform continuity/triangle-inequality propagation of local errors to multi-step closeness, and (iii) a compactness/Lebesgue-number type argument for neighborhoods of Γ_T in the product space. These match Lemmas 3.10–3.12 and Proposition 3.6 in the paper, and the final conversion between density-1 tracking and ρ_B=0 corresponds to Lemmas 2.2–2.3. Thus both are correct, and the proofs are substantially the same, differing only in metric presentation (sum-metric vs the paper’s ρ_∞) and minor normalization assumptions. Key citations: Theorem 3.13 and its proof (equivalence of pseudo-orbits) , , Proposition 3.6 (density characterization) , Lemmas 3.10–3.12 (density-of-blocks and neighborhood reduction) , , and Theorem 4.6 (VSP ⇔ AASP) , also highlighted as Theorem A in the introduction .