On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911691249287168 |
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| author | Anderson, Jack Boca, Florin P. Cobeli, Cristian Zaharescu, Alexandru |
| author_facet | Anderson, Jack Boca, Florin P. Cobeli, Cristian Zaharescu, Alexandru |
| contents | The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formula for their distribution in ${\mathcal V}$ as $Q\rightarrow \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07951 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions Anderson, Jack Boca, Florin P. Cobeli, Cristian Zaharescu, Alexandru Number Theory Primary: 11B57. Secondary: 11J71, 11K36, 11L05 The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formula for their distribution in ${\mathcal V}$ as $Q\rightarrow \infty$. |
| title | On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions |
| topic | Number Theory Primary: 11B57. Secondary: 11J71, 11K36, 11L05 |
| url | https://arxiv.org/abs/2508.07951 |