INTERIOR OF CERTAIN SUMS AND CONTINUOUS IMAGES OF VERY THIN CANTOR SETS

The paper’s Theorem 1.1 asserts that for any Cantor set K1 and any C^1 map H(α,x,y) with invertible Jacobian in the y-variable, there exists a Cantor set K2 and a parameter cube such that the intersection over all parameters of H(α,K1,K2) has nonempty interior (Theorem 1.1 and its restatement as Theorem 5.5) . However, taking H(α,x,y)=y satisfies the paper’s hypotheses (JH,2 is the identity), yet yields H(α,K1,K2)=K2 for all α, so the intersection is K2, which, being a Cantor set, has empty interior. Thus the theorem is false as stated. The proof contains a critical error: after using the implicit function theorem to define gc,α, the paper claims Jgc,α=JH,2^{-1}JH,1 is invertible “by our assumption” (that JH,2 is invertible) . This inference is invalid unless JH,1 is also invertible (or has full rank). In the counterexample H=y, JH,1=0 so Jgc,α=0, contradicting the claimed invertibility and breaking the subsequent argument. The model’s counterexample is therefore correct and exposes the flaw.