Improved dimension theory of sofic self-affine fractals

The paper’s main theorem (Theorem 2.2) states exactly the nested-sum formula for dim_H(K)=lim_{N→∞}(1/N)log_{m1} Z_N with a_i=log_{m_{i+1}}(m_i), together with the equivalence of the two displayed expressions; the proof proceeds via weighted topological entropy and a factor-chain argument, with the limit guaranteed by subadditivity in the appropriate covering counts . In contrast, the candidate solution attempts a direct covering/Frostman proof. Its submultiplicativity step (Fekete) and the equivalence of the two formulas are sound. However, the geometric covering and ball-measure estimates are flawed: (i) an N-cylinder image does not contain a sup-ball of radius ≍ m_1^{-N} (the inscribed sup-ball is controlled by the smallest side, ≍ m_r^{-N}); (ii) a sup-ball of radius ≍ m_1^{-N} intersects not a bounded number of N-cylinders but, in general, exponentially many in N, so the lower-bound/Frostman estimate as written fails. The paper’s approach avoids these pitfalls and supplies a complete proof via weighted entropy and variational principles .