On irrationals with Lagrange value exactly 3

For $c>0$, let $X_c$ denote the set of $x\in \mathbb{R}\mathrm{\backslash }\mathbb{Q}$ such that $\mid x-\frac{p}{q}\mid <\frac{1}{c{q}^2}$ 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 $n\in \mathbb{N}\cup \{\mathrm{\infty }\}$, the set of $x\in \mathbb{R}\mathrm{\backslash }\mathbb{Q}$ with Lagrange value exactly $3$ and such that $\mid x-\frac{p}{q}\mid <\frac{1}{3q^2}$ has exactly $n$ rational solutions $\frac{p}{q}$ is also uncountable.