Homomorphisms between multidimensional constant-shape substitutions

The paper’s Theorem 4.1 states exactly the claim: any measurable factor between two aperiodic primitive constant-shape substitutions with the same expansion L and support F1 is, up to a global shift, almost everywhere a sliding block code with explicit radius bound, and after passing to a power it intertwines the substitutions up to a fixed shift; see Theorem 4.1 and its bound and intertwining property (Sp ψ ζ1^n = ζ2^n ψ) in the text . The proof strategy in the paper uses recognizability and reducedness (definition of d_n and reducedness) and a “core with respect to a finite set C” provided by Proposition 2.10, alongside an a.e.-to-continuous upgrade via an iteration and finite-radius code selection argument (Lemmas 4.5–4.6; final steps quoted) . The candidate solution derives the same two conclusions and the same explicit bound, but with a different construction: it builds a quantitative ‘thick-core’ (ball-based) lemma, decodes letters using reducedness, applies Lusin plus CHL to get a global block code, and aligns phases to obtain the intertwining identity. Two minor issues: (i) their thick-core is phrased using Euclidean balls inside F_n, whereas the paper works with a combinatorial core defined via a fixed finite set C (Proposition 2.10), which avoids shape pathologies; replacing balls by the C-core fixes this without changing the bound or conclusions ; (ii) a brief claim that a certain a.e. equality set is closed uses continuity of φ, which holds only on the Lusin set—this is inessential since the theorem only needs a.e. equality. Overall, the results, radius bound, and intertwining statement match the paper’s theorem, but the proofs are methodologically different. A bijective special case discussed in the paper (radius 0 when property (2) holds) is also consistent with the candidate’s framework .