Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914014264557568 |
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| author | van Doorn, Wouter |
| author_facet | van Doorn, Wouter |
| contents | Let $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \ldots$ be the Farey fractions of order $n$. We then prove that the inequality $(a_l - a_k)(b_l - b_k) \ge 0$ holds for all $k$ and $l > k$ with $l-k \le \left(\frac{1}{12} - o(1) \right)n$, sharpening an old result by Erdős. On the other hand, we will show that for all $n \ge 4$ there are $k, l$ with $k < l < k + \frac{n}{4} + 5$ for which the product $(a_l - a_k)(b_l - b_k)$ is negative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions van Doorn, Wouter Number Theory Let $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \ldots$ be the Farey fractions of order $n$. We then prove that the inequality $(a_l - a_k)(b_l - b_k) \ge 0$ holds for all $k$ and $l > k$ with $l-k \le \left(\frac{1}{12} - o(1) \right)n$, sharpening an old result by Erdős. On the other hand, we will show that for all $n \ge 4$ there are $k, l$ with $k < l < k + \frac{n}{4} + 5$ for which the product $(a_l - a_k)(b_l - b_k)$ is negative. |
| title | Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.00121 |