Hausdorff measure of dominated planar self-affine sets with large dimension

The paper’s Theorem 1.1 states and proves the equivalence (a)⇔(b)⇔(c)⇔(d) for dominated planar self‑affine IFS with affinity dimension s0∈(1,2], using (i) a self‑affine ‘reverse slicing’ inequality (Lemma 3.1) to get (a)⇒(b), (ii) a Perron–Frobenius/transfer‑operator framework to obtain the dichotomy (b)⇔(c), and (iii) a representation of π∗μK via slice integrals (Proposition 2.4) to get (c)⇒(d), followed by the mass distribution principle for (d)⇒(a) (see Theorem 1.1 and its proof outline in the PDF, together with Lemma 3.1 and Propositions 2.1–2.4: ). By contrast, the candidate solution makes several overstrong and incorrect claims. Most notably, it asserts a uniform “thin‑rectangle” slicing lower bound c·∑|ι|=nφs0(Aι) ≤ ∫H∞s0−1(X∩proj−1V(t))dt which, together with the Gibbs property, would force the slice integral to be positive for every dominated system, contradicting examples with Hs0(X)=0 (e.g., Example 1.1 in the paper: ). It further claims a 1‑Frostman bound for (projV)∗π∗μK for all V∈XF without assuming Hs0(X)>0, whereas the paper shows this bound only as a consequence of Hs0(X)>0 (Theorem 1.2(i): and its proof ), and explicitly remarks on the difficulty of proving it in general (). The model’s Step 1 geometry is also misstated (axes “within a fixed angle of every V∈XF”). Therefore, the paper’s argument is correct and internally consistent, while the model’s solution contains substantive errors.