On General 2-dimensional Lattice Spectra: Closedness, Hall’s Ray, and Examples

The paper’s Theorem 49 states that for two general Cantor sets C,D⊂[0,1] (with endpoints 0 and 1) and a continuous, piecewise C^1 map g with positive partial derivatives, if the level-set slope ratio |∂g/∂α ÷ ∂g/∂β| lies in [Ap(C), Ap(C)^{-1}] ∩ [Ap(D), Ap(D)^{-1}] along g^{-1}(h), then G=g(C×D)=[g(0,0), g(1,1)] (statement and setup in 5.1–5.2 and Theorem 49) . The proof provided in the paper constructs nested rectangles B_s and performs a careful case analysis ensuring each B_s intersects the level set g^{-1}(h), and then refines to obtain a limiting rectangle inside C×D that still meets g^{-1}(h), completing the proof (inductive construction and case splits in the proof text, including the Q1–Q6 square analysis and the final refinement B_{∞,t} to force inclusion in C×D) . This approach is sound under the stated hypotheses. The candidate (model) solution proposes a different route via Newhouse thickness: it turns the level curve into a graph β=ψ_h(α), pulls D back to T_h=ψ_h^{-1}(D), estimates thickness distortion under a bi-Lipschitz map, and then invokes the Gap Lemma to force C∩T_h≠∅ for each h. However, the model relies on two incorrect or unjustified steps: (i) it asserts that restricting a Cantor set to a subinterval cannot decrease thickness, which is generally false; and (ii) it applies the Gap Lemma at the borderline τ(C)·τ(T_h)≥1 and assumes “linked position” from endpoint accumulation without justification. These gaps are not merely cosmetic, and they undermine the model’s proof. In contrast, the paper’s rectangle-based proof, though technical, supplies the needed inductive control to reach the conclusion.