On irrationals with Lagrange value exactly 3
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917258378346496 |
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| author | Cao, Zhe Erazo, Harold Moreira, Carlos Gustavo |
| author_facet | Cao, Zhe Erazo, Harold Moreira, Carlos Gustavo |
| contents | For $c>0$, let $X_c$ denote the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ such that $\left| x-\frac{p}{q} \right|<\frac{1}{cq^2}$ has only finitely many rational solutions $\frac{p}{q}$. It is a classical fact, known since the 1950s, that $X_c$ is uncountable for $c>3$ and countable for $c<3$. However, the cardinality of $X_3$ does not appear to be present in the literature. We prove that $X_3$ is uncountable.
More generally, we show that for any $n\in\mathbb{N}\cup\{\infty\}$, the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ with Lagrange value exactly $3$ and such that $\left| x-\frac{p}{q} \right|<\frac{1}{3q^2}$ has exactly $n$ rational solutions $\frac{p}{q}$ is also uncountable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On irrationals with Lagrange value exactly 3 Cao, Zhe Erazo, Harold Moreira, Carlos Gustavo Number Theory Combinatorics 11J06, 11J70 For $c>0$, let $X_c$ denote the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ such that $\left| x-\frac{p}{q} \right|<\frac{1}{cq^2}$ has only finitely many rational solutions $\frac{p}{q}$. It is a classical fact, known since the 1950s, that $X_c$ is uncountable for $c>3$ and countable for $c<3$. However, the cardinality of $X_3$ does not appear to be present in the literature. We prove that $X_3$ is uncountable. More generally, we show that for any $n\in\mathbb{N}\cup\{\infty\}$, the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ with Lagrange value exactly $3$ and such that $\left| x-\frac{p}{q} \right|<\frac{1}{3q^2}$ has exactly $n$ rational solutions $\frac{p}{q}$ is also uncountable. |
| title | On irrationals with Lagrange value exactly 3 |
| topic | Number Theory Combinatorics 11J06, 11J70 |
| url | https://arxiv.org/abs/2509.01867 |