A problem of Hirst for the Hurwitz continued fraction and the Hausdorff dimension of sets with restricted slowly growing digits

TL;DR

This work extends the dimension theory of continued fractions to complex Hurwitz expansions by proving that sets of complex irrationals with digits drawn from an infinite Gaussian-integer subset $S$ and with digits diverging in size have Hausdorff dimension $\frac{\tau(|S|)}{2}$, where $\tau(|S|)$ is the convergence exponent of $|S|$. The authors develop and apply a two-pronged framework: (i) a 2-decaying conformal IFS construction associated with Hurwitz digits to obtain precise upper bounds, and (ii) a non-autonomous conformal IFS approach together with Bowen's formula to obtain matching lower bounds. This yields $\dim_H F(S)=\dim_H F(S,f)=\frac{\tau(|S|)}{2}$ for any infinite $S\subset\mathbb Z(i)$ and slowly growing restrictions $f(n)\to\infty$, thereby confirming Hirst's conjecture in the Hurwitz complex setting and providing a complex-analytic analogue of prior real-case results. A key corollary shows that $\dim_H\{z:|c_n(z)|\to\infty\}=1$ for the Hurwitz expansion, with the convergence exponent $\tau(|\mathbb Z(i)|)=2$, highlighting a sharp fractal dimension transition induced by digit growth in complex continued fractions.

Abstract

We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we confirm Hirst's conjecture, as a complex analogue of the result of Wang and Wu [Bull. Lond. Math. Soc. {\bf 40} (2008), no. 1, 18--22] for the regular continued fraction. We also prove a complex analogue of the second-named author's result on the Hausdorff dimension of sets with restricted slowly growing digits [Proc. Amer. Math. Soc. {\bf 151} (2023), no. 9, 3645--3653]. To these ends, we exploit an infinite conformal iterated function system associated with the Hurwitz continued fraction.

Figures (2)

• Figure 1: The domain $U_{k,\ell}$ ($(k,\ell)\in\mathbb D_1$) is bordered by the four circles through the origin, orthogonally intersecting each other.
• Figure 2: The collection $\{[c_1]\colon c_1=(k,\ell)\in\mathbb D_1 \}$ of $1$-cylinders tessellates $U$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2: Ger
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5