Truncated Hurwitz Continued Fractions

• Truncated Hurwitz continued fraction expansion is a method that generalizes regular continued fractions to complex fields, offering finite, algebraically explicit approximations for quadratic units.
• It employs a nearest-integer algorithm within fundamental domains (square for Gaussian and hexagonal for Eisenstein integers) to achieve double-exponential convergence with explicit error bounds.
• The technique reveals deep dynamical and ergodic properties, linking iterative methods, symbolic dynamics, and Diophantine approximation in complex quadratic fields.

The truncated Hurwitz continued fraction expansion generalizes the concept of regular continued fractions to the setting of complex numbers, particularly over imaginary quadratic number fields, and provides strong algebraic, analytic, and dynamical links to classical and modern number theory. The notion of truncation refers to the finite continued fraction expansion obtained at a prescribed step in the Hurwitz algorithm, with the expansion's structure deeply reflecting the arithmetic and dynamical properties of the underlying number field and the type of number being approximated (e.g., quadratic units, transcendental numbers).

1. Algorithmic Structure and Fundamental Domains

The Hurwitz continued fraction expansion utilizes a nearest-integer algorithm tailored to the lattice structure of the underlying field. For the Gaussian integers $\mathbb{Z}[i]$ (or Eisenstein integers $\mathfrak o(\sqrt{-3})$), a complex number $\alpha$ is first normalized to lie in a fundamental domain $X$ (typically square or hexagonal). The partial quotients $a_n$ are computed as

$a_n = a_H(\alpha^{(n)}),$

where $\alpha^{(1)} = \alpha$ and $\alpha^{(n+1)} = T_H(\alpha^{(n)})$. The Hurwitz transformation $T_H$ is given by

$$T_H(z) = \begin{cases} (z - \lfloor z \rfloor_H)^{-1} & \text{if } z \neq 0, \ 0 & \text{if } z = 0, \end{cases}$$

with $\lfloor z \rfloor_H$ denoting the nearest lattice point in $(1 + i)\mathbb{Z}[i]$ (resp., $\eta \cdot \mathfrak o(\sqrt{-3})$ in the Eisenstein case) (Yasutomi, 22 Oct 2024, Hitoshi et al., 20 Mar 2025). Truncation occurs naturally when $T_H(\alpha^{(k)}) = 0$ after $k$ steps, yielding a finite expansion,

$\alpha = [0; a_1, a_2, \ldots, a_k].$

For irrational (esp. quadratic irrational) inputs, the expansion is either infinite or eventually periodic, and truncated expansions (convergents) provide rational approximants.

2. Explicit Formulas for Truncated Expansions of Quadratic Units

Quadratic units—roots of $f(X) = X^2 - tX + u$ with $t$ in the corresponding integer ring and $u$ a suitable unit—possess highly regular Hurwitz expansions. For instance, in the Gaussian case, the expansion for $\alpha$ (root with $|\alpha| < 1/2$) is (Saito et al., 17 Oct 2025): $\alpha = [0; u^{-1}t, -t, u^{-1}t, -t, \ldots],$ with the sequence alternating between $u^{-1}t$ and $-t$. The length of the truncated expansion corresponding to the $n$th iterate is $2^{n+1}-1$.

The value of the truncated Hurwitz expansion of length $2^{n+1}-1$ is exactly given by the $(n+1)$-fold Newton approximation $F^{(n+1)}(0)$ for $f(X)$, and by the truncated Sierpiński series $S_n(t,u)$ (Saito et al., 17 Oct 2025): $$F^{(n+1)}(0) = S_n(t, u) = [0; a_1, a_2, ..., a_{2^{n+1}-1}],$$ where

$$S_n(T,U) = \sum_{m=0}^n \frac{U^{2^m}}{h_0(T) h_1(T,U) \cdots h_m(T,U)},\quad h_0 = T,\quad h_{m+1} = h_m^2 - 2U^{2^m}.$$

This equivalence provides a uniform explicit description of the truncated approximants for quadratic units, bridging iterative, series, and continued fraction perspectives.

3. Properties of Convergents and Error Bounds

The truncated Hurwitz expansion's $n$th convergent $C_n = [0; a_1, ..., a_n]$ yields an efficient rational approximation to $\alpha$. The paper establishes rapid convergence: $$| \alpha - F^{(n)}(0) | < \frac{2}{\rho^{2^{n+1}-1} },\quad \rho = \frac{ |t| + \sqrt{ |t|^2 - 4 } }{2 } > 1,$$ for any $n \geq 1$. This double-exponential decay in error with respect to $n$ arises from the recurrence's structure and the growth properties of the partial denominators. Comparable monotonicity results are provided for the Eisenstein-field case, where the principal denominator $|q_n|$ increases strictly with $n$ (Hitoshi et al., 20 Mar 2025).

Such bounds decisively outperform classical continued fraction convergents associated with regular algorithms, attesting to the strength of truncated Hurwitz expansions for quadratic units.

4. Dynamical and Ergodic Features

The Hurwitz continued fraction map on the complex plane admits a natural extension with an ergodic absolutely continuous invariant measure. For the transformation $T_H$, the invariant density $h(z, w) = C |z-w|^{-4}$ ensures statistical regularity of digit distribution in the expansion (Yasutomi, 22 Oct 2024). In the Eisenstein module setting, $T$ also possesses a finite-range structure and ergodic invariant probability measure (Hitoshi et al., 20 Mar 2025).

These dynamical properties facilitate the paper of metric Diophantine approximation in complex quadratic fields and underpin applications in normal numbers and descriptive set theory (García-Ramos et al., 2023).

5. Applications and Connections

The truncated Hurwitz continued fraction expansion finds application across several domains:

• Diophantine approximation: Truncated convergents deliver sharp rational approximations to algebraic and transcendental numbers in quadratic fields, with error rates tied to the recurrence structure.
• Symbolic dynamics: The subshift space of Hurwitz expansions, particularly the “regularized” closure, exhibits specification and concatenation properties relevant for descriptive set theory.
• Algorithmic computation: For rational or quadratic irrational inputs, truncation yields explicit finite expansions, facilitating arithmetic representation, root finding, and numerical algorithms in the complex setting.
• Iterative methods: The paper demonstrates the equivalence between truncated Hurwitz expansions, Newton iterations, and Sierpiński series in the computation of quadratic units, supporting iterative analytic approaches to algebraic numbers (Saito et al., 17 Oct 2025).

6. Comparison with Classical and Alternative Expansions

While the Hurwitz expansion generalizes the regular continued fraction theory to complex numbers, its dynamical, combinatorial, and approximative properties differ fundamentally:

• The truncated Hurwitz expansion for rationals terminates, analogous to truncation in the real classical case (Yasutomi, 22 Oct 2024).
• For irrationals (and particularly quadratic units), Hurwitz expansions are either eventually or purely periodic, echoing the classical Lagrange theorem (Yasutomi, 22 Oct 2024).
• Unlike real continued fractions—where the expansion of a close rational approximant agrees in its prefix with that of the target number—the complex Hurwitz expansions may diverge dramatically in their digit sequences even for extremely good rational approximants (He et al., 2021). This fundamental feature is quantified dimensionally by showing that the set of numbers with maximally differing truncated Hurwitz expansions from their approximants has full packing dimension.

7. Generalizations and Extensions

Recent developments include the definition and analysis of Hurwitzian expansions indexed by modules over other quadratic fields, e.g., Eisenstein integers, with hexagonal fundamental domains and tailored nearest-integer functions (Hitoshi et al., 20 Mar 2025). Explicit truncated expansions and error estimates are also available, using carefully regularized expansions to ensure convergence and monotonic growth in denominators.

Table: Characteristic Features of Truncated Hurwitz Expansions

Domain	Expansion Algorithm	Convergence Rate / Error Estimate
Gaussian	Nearest-$\mathbb{Z}[i]$ via square domain	$2/\rho^{2^{n+1}-1}$ (double-exponential)
Eisenstein	Nearest-$\mathfrak o(\sqrt{-3})$ via hexagon	Strict monotonic increase in $|q_n|$; similar error rate
Classical real	Floor function, linear recurrence	Exponential rate, polynomial error bound

Conclusion

The truncated Hurwitz continued fraction expansion manifests as a highly structured, rapidly convergent, and algebraically explicit form for approximating quadratic units and general complex numbers in imaginary quadratic fields. Its tight links to iterative methods, explicit series, and ergodic theory underscore its broad applicability and centrality in the modern paper of complex continued fractions, Diophantine approximation, and symbolic dynamics. The unique phenomena emerging in truncation—such as the divergence of expansions for close approximants, the specification property in symbolic sequence space, and the equivalence between different analytic representations—continue to motivate deeper exploration into the interplay between algebraic and dynamical aspects of continued fraction theory.