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 <italic>finite range condition</italic> plays a prominent role. We show that this condition holds for complex <italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic alpha"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic"> α </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\boldsymbol {\alpha }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Hurwitz algorithms </italic> with parameters <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic alpha element-of double-struck upper Q squared"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic"> α </mml:mi> </mml:mrow> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\boldsymbol {\alpha }}\in \mathbb {Q}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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].