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arXiv · 2510.17112

Representations of rational numbers and Minkowski dimension

Haipeng Chen·Lai Jiang·Yufeng Wu
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Abstract

In this paper, we investigate the representations of rational numbers via continued fraction, Egyptian fraction, and Engel fraction expansions. Given $m \in \mathbb{N}$, denote by $C_m, E_m, E_m^*$ the sets of rational numbers whose continued fraction, Egyptian fraction, and Engel fraction expansions have length $m$, respectively. We first establish the Minkowski dimensions of these sets, which implies that their global scaling properties are different. We also apply the results to sumsets of decreasing sequences.

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Haipeng Chen, Lai Jiang, Yufeng Wu. 2025-10-20. Representations of rational numbers and Minkowski dimension. https://arxiv.org/abs/2510.17112

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