Exact Hausdorff dimension of some sofic self-affine fractals

The paper’s planar main theorem (Theorem 3.2) assumes primitivity of A=∑i Ai and a synchronizing word s with ImR(As)=Span{v⊤}, defines Ju by v⊤AuAs=Juv⊤ and Ck:=∑|u|=k,u avoids s Jαu, and concludes dimH(X)=logm1 r where r>0 uniquely solves rL=∑k≥0 Ck r−k; this is proved via a tower decomposition, a countable operator M, and Perron–Frobenius arguments (ΦN+1=MΦN and spectral radius) . The candidate solution reaches the same formula using a renewal/generating-function approach F(z)=zL H(z)/(1−zL C(z)) and a primitivity-based ‘bridging lemma’ to compare restricted and full partition sums. Differences are technical rather than substantive: the paper fixes the entrywise ℓ1-matrix norm in §3.1 (while noting earlier that any norm can be used in the dimension limit) , whereas the candidate uses the operator 1-norm; these norms are equivalent for nonnegative matrices, so the exponential growth (and hence the dimension) is unchanged. The candidate’s existence/uniqueness of r is best justified via the paper’s spectral-radius argument (or monotonicity in r) rather than via potential blow-up at the radius of convergence. Overall, both arguments are correct and produce the same result, via different presentations.