  We investigate the averages of Dedekind sums over rational numbers in the set
  $$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0,
\alpha)$$
  for fixed $\alpha\leq 1/2$.
  In previous work, we obtained asymptotics for $\alpha=1/2$, confirming a
conjecture of Ito in a quantitative form.
  In the present article we extend our former results, first to all fixed
rational $\alpha$ and then to almost all irrational $\alpha$.
  As an intermediate step we obtain a result quantifying the bias occurring in
the second term of the asymptotic for the average running time of the
\textit{by-excess} Euclidean algorithm, which is of independent interest.
