Suppose $k_n$ denotes either $\phi
(n)$ or $\phi (p_n)$ $(n \ = \ 1,2,\cdots )$ where the
polynomial $\phi$ maps the natural numbers to themselves
and $p_k$ denotes the $k^{th}$ rational prime. Let
$(\frac{r_n}{q_n})_{n=1}^{\infty}$ denote the sequence of
convergents to a real number $x$ and define the the
sequence of approximation constants $(\theta
_n(x))_{n=1}^{\infty}$ by
$$
\theta _n(x) \ = \ q_n^2\left\vert x \ - \ \frac{r_n}{q_n}\right\vert
.
\eqno (n \ = \ 1,2, \cdots )
$$
In this paper we study the behaviour of the sequence
$(\theta _{k_n}(x))_{n=1}^{\infty}$ for almost all $x$
with respect to Lebesgue measure. In the special case
where $k_n \ = \ n$ $(n \ = \ 1,2,\cdots )$ these
results are due to W. Bosma, H. Jager and F. Wiedijk.