On irrationals with Lagrange value exactly 3

Abstract

For $c>0$, let $X_c$ denote the set of $x\in \mathbb{R}\backslash \mathbb{Q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{c{q}^2}$ 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.

After publication, we learned that the cardinality of $X_3$ was obtained by C. Gurwood (PhD thesis, 1976) by fully characterizing $X(0)$. The result was later reproved by a different method by G. Harcos (undergraduate thesis, 1996, Hungarian). Our method allows the construction of uncountably many irrational numbers in each X(n) with Sturmian continued fraction after substituting a=(2,2), b=(1,1)