On restricted averages of Dedekind sums

We investigate the averages of Dedekind sums over rational numbers in the set $${\mathcal{F}}_{\alpha }(Q):=\{v\mathrm{/}w\in \mathbb{Q}:0<w\le Q\}\cap [0,\alpha )$$ for fixed $\alpha \le 1\mathrm{/}2$. In previous work, we obtained asymptotics for $\alpha =1\mathrm{/}2$, 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 $\mathit{\text{by-excess}}$ Euclidean algorithm, which is of independent interest.