A finiteness condition for complex continued fraction algorithms

It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show that this condition holds for complex α-Hurwitz algorithms with parameters α ∈ Q2. This is equivalent to the existence of certain partitions with finitely many atoms related to these algorithms and lies at the root of explorations into their Diophantine properties. Our result provides a partial answer to a recent question formulated by Lukyanenko and Vandehey [Conform. Geom. Dyn. 29 (2025), pp. 57–89].