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arXiv · 2602.09700

On very badly approximable numbers

Zhe Cao·Harold Erazo·Carlos Gustavo Moreira
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Abstract

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 $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$.

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Zhe Cao, Harold Erazo, Carlos Gustavo Moreira. 2026-02-10. On very badly approximable numbers. https://arxiv.org/abs/2602.09700

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