Representations of rational numbers and Minkowski dimension
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918262638379008 |
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| author | Chen, Haipeng Jiang, Lai Wu, Yufeng |
| author_facet | Chen, Haipeng Jiang, Lai Wu, Yufeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17112 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representations of rational numbers and Minkowski dimension Chen, Haipeng Jiang, Lai Wu, Yufeng Classical Analysis and ODEs Number Theory 28A80, 11A55, 11A67 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. |
| title | Representations of rational numbers and Minkowski dimension |
| topic | Classical Analysis and ODEs Number Theory 28A80, 11A55, 11A67 |
| url | https://arxiv.org/abs/2510.17112 |