TYPICAL SELF-AFFINE SETS WITH NON-EMPTY INTERIOR

The paper proves Theorem 1.1 under the small-norm hypothesis ||Ti||<1/2 by: (i) constructing a measure μ on Σ with the one-sided cylinder bound μ([I]) ≤ C g_t(I) r^{|I|} for g_t(I)=α_d(T_I)^t|det T_I| (Lemma 4.2), (ii) establishing an oscillatory-integral bound averaged over parameters a (Proposition 3.3), and (iii) converting it, after integrating in ξ with a precise kernel estimate (Proposition 3.4), into the finiteness of ∫_{B(0,ρ)}∫ |μ̂_a(ξ)|^2 ||ξ||^t dξ da (equation (4.8)), whence dim_S μ_a ≥ t+d > 2d and thus Ka has non-empty interior by Lemma 2.2. These steps and thresholds are explicit in the text (Theorem 1.1, Lemma 4.2, Propositions 3.3–3.4, Lemma 2.2) . By contrast, the candidate solution asserts two claims that are not supported by the paper’s arguments and are, in fact, incorrect as stated: (1) a two-sided quasi-Bernoulli comparability for cylinder weights ν([I]) ≍ α_d(T_I)^s |det T_I|, which does not follow from Lemma 4.2 (the paper only proves an upper bound μ([I]) ≤ C g_t(I) r^{|I|}); and (2) an averaged pointwise Fourier decay ∫_B |μ̂_a(ξ)|^2 da ≲ (1+|ξ|)^{-s}, which the paper does not establish (the paper proves only an integrated estimate in ξ with weight ||ξ||^t). Most critically, the candidate misapplies Mattila’s criterion by using s>d to conclude a continuous density; the correct threshold, as used in the paper, is dim_S μ > 2d (equivalently, an energy with exponent >2d), not merely >d (Lemma 2.2) . Hence the model’s argument fails to reach the required absolute continuity/continuity threshold and does not justify non-empty interior.