Hurwitz's theorem for Diophantine approximation

Question

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, there are infinitely many relatively prime natural numbers $(p,q)$ satisfying the equation: $$ | \xi−\frac{p}{q}| < \frac{1}{\sqrt{5}q^2} $$

I'm interested in unilateral approximations to $\xi.$ Specifically, letting

$$ L = \left\{(p,q): \; p \text{ and } q \text{ are relatively prime positive integers such that } \frac{p}{q} < \xi \text{ and } \left|\xi - \frac{p}{q}\right| < \frac{1}{\sqrt{5}q^2} \right\} $$

and

$$ U = \left\{(p,q): \; p \text{ and } q \text{ are relatively prime positive integers such that } \frac{p}{q} > \xi \text{ and } \left|\xi - \frac{p}{q}\right| < \frac{1}{\sqrt{5}q^2} \right\}, $$

then for each irrational $\xi$ we can conclude that AT LEAST ONE of the sets $L$ and $U$ is infinite. Do we know whether, for each irrational $\xi,$ BOTH of the sets are infinite? If not, then do we at least know whether, for each irrational $\xi,$ BOTH of the sets are not empty?

Answer 1

Theorem 1 of the Eggan & Niven paper mentioned in the comments says,

For any $c>1$ there are uncountably many irrationals $\xi$ for which $$0<{a\over b}-\xi<{1\over cb^2}$$ has no rational solutions $a/b$. Analogously, for any $c>1$ there are uncountably many irrationals $\xi$ for which $0<\xi-(a/b)<1/(cb^2)$ has no rational solutions.