In this note the distribution of the approximation coefficients
$\Theta_n$,
associated with the regular continued fraction expansion of numbers
$x\in [0,1)$, is given under extra conditions on the numerators and
denominators of the convergents $p_n/q_n$. Similar results are also
obtained for $S$-expansions. Further, a Gauss-Kusmin type theorem is
derived for the regular continued fraction expansion under these extra
conditions.