Minkowski dimension and content of complex continued fractions

2026/07/21 by Yifei Gu, Lai Jiang
#math.CA #math.NT

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Abstract

We study the Minkowski geometry of finite-level sets of Gaussian rationals arising from Hurwitz continued fractions. For each m≥ 1, let H_m be the set of points in the fundamental square whose Hurwitz continued fraction expansions have length exactly m. We also consider the relaxed recursive sets defined by G₀=\0\ and G_m=\(1)/(u+v): u ∈ℤ[i], v∈ G_m-1, |u+v|>1 \. We prove that for every m≥ 1, dim_{\rm M} H_m=dim_{\rm M} G_m=1. We further determine the critical one-dimensional Minkowski content of these sets. We have \mathcal M¹(H₁)=\mathcal M¹(G₁)=4πlog(1+√(2)), whereas \mathcal M¹(H_m)=\mathcal M¹(G_m)=∞ for every m≥ 2.

Citations

• A problem of Hirst on continued fractions with sequences of partial quotients
• Fractal Geometry: Mathematical Foundations and Applications.
• A problem of Hirst for the Hurwitz continued fraction and the Hausdorff dimension of sets with restricted slowly growing digits

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• Metrical properties of Hurwitz Continued Fractions
• A problem of Hirst for the Hurwitz continued fraction and the Hausdorff dimension of sets with restricted slowly growing digits
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