Distance between cubics and rationals
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917181962321920 |
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| author | Badziahin, Dmitry |
| author_facet | Badziahin, Dmitry |
| contents | We investigate the following problem: what is the smallest possible distance between a cubic irrational $ξ$ and a rational number $p/q$ in terms of the height $H(ξ)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of all pairs $(u,v)$ of positive real numbers such that $|ξ- p/q| > cH^{-u}(ξ)q^{-v}$ for all cubic irrationals $ξ$ and rationals $p/q$. First, we transform this problem into one about the root separation of cubic polynomials. Second, under the assumption of the famous abc-conjecture, we give an almost complete description of $D_{3,1}$. Namely, the points $(u,v)$ with $2\le v\le 3$ that lie in the interior of $D_{3,1}$ are characterised by the inequality $u> 10-3v$. Assuming only the weaker Hall conjecture, we also obtain nontrivial results about the shape of $D_{3,1}$, although these are not as strong as those derived from the abc-conjecture. Finally, we discuss an analogue of the set $D_{3,1}$ in function fields where we are able to give an almost complete description unconditionally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01105 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distance between cubics and rationals Badziahin, Dmitry Number Theory 11J68, 11D59 We investigate the following problem: what is the smallest possible distance between a cubic irrational $ξ$ and a rational number $p/q$ in terms of the height $H(ξ)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of all pairs $(u,v)$ of positive real numbers such that $|ξ- p/q| > cH^{-u}(ξ)q^{-v}$ for all cubic irrationals $ξ$ and rationals $p/q$. First, we transform this problem into one about the root separation of cubic polynomials. Second, under the assumption of the famous abc-conjecture, we give an almost complete description of $D_{3,1}$. Namely, the points $(u,v)$ with $2\le v\le 3$ that lie in the interior of $D_{3,1}$ are characterised by the inequality $u> 10-3v$. Assuming only the weaker Hall conjecture, we also obtain nontrivial results about the shape of $D_{3,1}$, although these are not as strong as those derived from the abc-conjecture. Finally, we discuss an analogue of the set $D_{3,1}$ in function fields where we are able to give an almost complete description unconditionally. |
| title | Distance between cubics and rationals |
| topic | Number Theory 11J68, 11D59 |
| url | https://arxiv.org/abs/2509.01105 |