Metrical properties of Hurwitz continued fractions

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics Pub Date : 2025-03-18 DOI:10.1016/j.aim.2025.110208

Yann Bugeaud , Gerardo González Robert , Mumtaz Hussain

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引用次数: 0

Abstract

We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.

Let

Φ : N \to R_{> 0}

be any function. For any complex number z and

n \in N

, let

a_{n} (z)

denote the nth partial quotient in the Hurwitz continued fraction of z. One of the main results of this paper is the computation of the Hausdorff dimension of the set

E (Φ) : = {z \in C : | a_{n} (z) | \geq Φ (n) for infinitely many n \in N} .

This study is a complex analog of a well-known result of Wang and Wu (2008) [55].

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来源期刊

Advances in Mathematics 数学-数学

CiteScore

2.80

自引率

5.90%

发文量

497

审稿时长

7.5 months

期刊介绍： Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.