Transcendence and normality of complex numbers via Hurwitz continued fractions

TL;DR

The paper analyzes Hurwitz continued fractions in the complex plane, showing that the natural sequence space is not closed and introducing a canonical closed subset $ar{oldsymbol{R}}$ that, via a continuous map $ar{oldsymbol{ extLambda}}$, encodes all complex numbers outside $oldsymbol{Q}(i)$. It proves that $(ar{oldsymbol{R}},oldsymbol{ extsigma})$ is a subshift with a feeble specification property and uses this structure to establish that Hurwitz normal numbers form a $oldsymbol{ extPi}^0_3$-complete set in the Borel hierarchy. The work also constructs a family of transcendental complex numbers with bounded partial quotients by combining bounded repetition exponents with Schmidt’s Subspace Theorem, illustrating nontrivial transcendence phenomena beyond prior criteria. Overall, the results connect complex dynamical systems, descriptive set theory, and number theory, yielding both structural insights and explicit transcendental examples.

Abstract

We study the topological, dynamical, and descriptive set theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian rational. The resulting space of sequences of Gaussian integers $Ω$ is not closed. By means of an algorithm, we show that $Ω$ contains a natural subset whose closure $\overline{\mathsf{R}}$ encodes continued fraction expansions of complex numbers which are not Gaussian rationals. We prove that $(\overline{\mathsf{R}}, σ)$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

Figures (8)

• Figure 1: From left to right: $\mathfrak{F}^{\circ}$, $\mathfrak{F}^{\circ}_1(-2)$, $\mathfrak{F}^{\circ}_1(-2+i)$, $\mathfrak{F}^{\circ}_1(-1+i)$.
• Figure 2: The cylinders $\mathcal{C}_1(a)$, $a\in\mathscr{D}$, and the numbers $\zeta_1,\zeta_2,\zeta_3,\zeta_4$.
• Figure 3: The set $\iota[\mathfrak{F}^{\circ}]$.
• Figure 4: The sets $\mathfrak{F}^{\circ}\setminus \overline{\mathbb{D}}(1+i)$ (left) and $\iota[\overline{\mathfrak{F}}\setminus \mathbb{D}(1+i)]$ (right).
• Figure 5: The sets $\mathfrak{F}^{\circ}\setminus \overline{\mathbb{D}}(-1)$ (left) and $\iota[ \overline{\mathfrak{F}}\setminus \mathbb{D}(-1)]$ (right).

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3: BugGeroHus2023
• Proposition 3.4
• Lemma 3.5
• proof