On the asymptotic behavior of Dedekind sums

Abstract

Let $z$ be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums $S(p_k,q_k)$ belonging to convergents $p_k/q_k$ of the {\em regular} continued fraction expansion of $z$ with that of Dedekind sums $S(s_j/t_j)$ belonging to convergents $s_j/t_j$ of the {\em negative regular} continued fraction expansion of $z$. Whereas the three main cases of this behavior are closely related, a more detailed study of the most interesting case (in which the Dedekind sums remain bounded) exhibits some marked differences, since the cluster points depend on the respective periods of these expansions. We show in which cases cluster points of $S(s_j,t_j)$ can coincide with cluster points of $S(p_k,q_k)$. An important tool for our purpose is a criterion that says which convergents $s_j/t_j$ of $z$ are convergents $p_k/q_k$.