Dedekind sums in the vicinity of quadratic irrationals

We study the behaviour of the classical Dedekind sums s(m/n) for convergents m/n of a given quadratic irrational α. It turns out that two cases may occur: Either the sequence s(m/n) remains bounded with finitely many cluster points, or s(m/n) tends to +∞ or −∞ like or , respectively. By means of the Barkan–Hickerson–Knuth formula we obtain a precise description of what happens in all cases.