A FINITENESS CONDITION FOR COMPLEX CONTINUED FRACTION ALGORITHMS

The paper establishes the finite range condition for α-Hurwitz algorithms with rational parameters α=(p/q,r/s)∈D by (i) reducing the problem to bounding the forward images of the boundary via Lemma 4.1, and (ii) proving finiteness of a certain family of generalized circles through explicit inversion–translation recurrences, which yields a finite partition and hence the finite range property . By contrast, the candidate solution’s Step 4 asserts the identity Tα(Δα(b1,…,bn)) = ⋂j Tα(Δα(bj)), which is false (already for n=2 one has Tα(Δα(b1,b2)) = Tα(Δα(b1)) ∩ Δα(b2), not ∩j Tα(Δα(bj))), and the claimed generality “for all α∈D” contradicts the discussion that for some irrational parameters the finite partition property fails .