On very badly approximable numbers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911437636501504 |
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| author | Cao, Zhe Erazo, Harold Moreira, Carlos Gustavo |
| author_facet | Cao, Zhe Erazo, Harold Moreira, Carlos Gustavo |
| contents | We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational solutions: their continued fraction is eventually a balanced sequence through a simple coding. As consequence, we show that all such numbers are either quadratic surds or transcendental numbers. In particular, for any algebraic real number $x$ of degree at least $3$ there are infinitely rational numbers $\frac{p}{q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09700 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On very badly approximable numbers Cao, Zhe Erazo, Harold Moreira, Carlos Gustavo Number Theory Combinatorics 11J06, 11J70, 68R15 We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational solutions: their continued fraction is eventually a balanced sequence through a simple coding. As consequence, we show that all such numbers are either quadratic surds or transcendental numbers. In particular, for any algebraic real number $x$ of degree at least $3$ there are infinitely rational numbers $\frac{p}{q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$. |
| title | On very badly approximable numbers |
| topic | Number Theory Combinatorics 11J06, 11J70, 68R15 |
| url | https://arxiv.org/abs/2602.09700 |