  We study the asymptotic behaviour of the classical Dedekind sums $s(s_k/t_k)$
for the sequence of convergents $s_k/t_k$ $k\ge 0$, of the transcendental
number \BD
  \sum_{j=0}^\infty\frac {1}{b^{2^j}},\ b\ge 3. \ED In particular, we show that
there are infinitely many open intervals of constant length such that the
sequence $s(s_k/t_k)$ has infinitely many transcendental cluster points in each
interval.
