  The paper deals with best one--sided (lower or upper) Diophantine
approximations of the $\ell$-th kind ($\ell\in\mathbb{N}$). We use the ordinary
continued fraction expansions to formulate explicit criteria for a fraction
$\frac{p}{q}\in\mathbb{Q}$ to be a best lower or upper Diophantine
approximation of the $\ell$-th kind to a given $\alpha\in\mathbb{R}$. The sets
of best lower and upper approximations are examined in terms of their
cardinalities and metric properties. Applying our results in spectral analysis,
we obtain an explanation for the rarity of so-called Bethe--Sommerfeld quantum
graphs.
