On Arithmetic Sums of Fractal Sets in R^d

The paper proves that uniformly non-flat (positive-thickness) sets are arithmetically thick and then establishes arithmetic thickness for self-affine attractors under (i) commuting linear parts, (ii) irreducibility plus a proximal element, and (iii) all planar cases; see the definitions, Theorem 1.2, and Theorem 1.6 in the paper . Crucially, the authors emphasize that in the settings of Theorems 1.4–1.6 a self-conformal/self-affine set not in a hyperplane may still have zero thickness, so one cannot prove Theorems 1.4–1.6 merely by verifying uniform non-flatness of E and invoking Theorem 1.2 . Their proofs for (ii) and (iii) proceed via microsets and span arguments (Propositions 6.3–6.4) and a planar cone/Perron–Frobenius analysis, not by showing E is uniformly non-flat . The candidate solution incorrectly asserts that under (i)–(iii) one can verify uniform non-flatness of E and then apply the general criterion; this is explicitly contradicted by the paper’s remark that E may have zero thickness in these settings .