Zaremba’s conjecture over imaginary quadratic fields

Establish that for each N ≥ 1 and for an imaginary quadratic field ℚ(√−d) with ring of integers O_d, there exist α, β ∈ O_d with ht(α/β)^2 = N such that the complex continued fraction expansion of α/β (defined via the nearest-integer complex Gauss map on the fundamental domain) has all digits α_j in a fixed bounded set A_d ⊂ O_d.

Background

The paper extends the classical setting to complex continued fractions over imaginary quadratic fields, using the Gauss-type map T_d defined on a fundamental domain compatible with the lattice structure of O_d.

A height function ht(z) = max{|α|, |β|} is used to parametrize rationals z = α/β ∈ ℚ(√−d). The conjecture seeks bounded partial quotients (from a bounded subset A_d of O_d) in the complex continued fraction expansion for every prescribed height level, mirroring Zaremba’s original conjecture in the complex setting.

References

It is then also natural to propose the following version of complex Zaremba conjecture; See also relevant contexts, e.g. for the case $d=1$. Let $N \in $. Then there exist $\alpha, \beta \in \mathcal{O}d$ with $\mathrm{ht}(\alpha/\beta)2=N$ such that \frac{\alpha}{\beta}=[0; \alpha_1, \alpha_2, \ldots, \alpha\ell] has all $\alpha_j \in \mathcal{A}_d$ for some bounded set $\mathcal{A}_d \subset \mathcal{O}_d$.

— Asymptotic statistics for finite continued fractions with restricted digits (2512.11357 - Lee, 12 Dec 2025) in Section 1 (Introduction), Conjecture [Zaremba over imaginary quadratic fields]