On very badly approximable numbers

TL;DR

This paper characterizes certain badly approximable irrationals with specific continued fraction properties, showing they are either quadratic surds or transcendental, and establishes infinite approximations for algebraic numbers of degree at least 3.

Contribution

It provides a complete characterization of irrationals with finitely many close rational approximations under a specific bound, linking their continued fractions to algebraic and transcendental classifications.

Findings

Numbers with finitely many close rational approximations are either quadratic surds or transcendental.

Continued fractions of these numbers are eventually balanced sequences.

Algebraic numbers of degree ≥ 3 have infinitely many close rational approximations.

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