Möbius inversion and coprime summation for error-sum functions of continued fractions
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arXiv
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| Format: | Preprint |
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2025
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| author | Ahn, Min Woong |
| author_facet | Ahn, Min Woong |
| contents | We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff dimension of the graph of $\mathcal{E}$ is exactly equal to $1$. Our proof is number-theoretic in nature and involves Möbius inversion, summation over coprime convergent denominators, and precise upper bounds derived via continued fraction recurrence relations. As a supplementary result, we rederive the known upper bound of $3/2$ for the Hausdorff dimension of the graph of the relative error-sum function $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Möbius inversion and coprime summation for error-sum functions of continued fractions Ahn, Min Woong Number Theory Classical Analysis and ODEs Primary 11A55, 11K50, Secondary 26A18, 28A80, 37E05, 33E20 We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff dimension of the graph of $\mathcal{E}$ is exactly equal to $1$. Our proof is number-theoretic in nature and involves Möbius inversion, summation over coprime convergent denominators, and precise upper bounds derived via continued fraction recurrence relations. As a supplementary result, we rederive the known upper bound of $3/2$ for the Hausdorff dimension of the graph of the relative error-sum function $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$. |
| title | Möbius inversion and coprime summation for error-sum functions of continued fractions |
| topic | Number Theory Classical Analysis and ODEs Primary 11A55, 11K50, Secondary 26A18, 28A80, 37E05, 33E20 |
| url | https://arxiv.org/abs/2507.16536 |