On quadratic irrationals with bounded partial quotients

Abstract

It is shown that for some explicit constants $$c>0, A>0$$ , the asymptotic for the number of positive non-square discriminants $$D<x$$ with fundamental solution $$\varepsilon _D<x^{\frac{1}{2}+\alpha }$$ , $$0<\alpha <c$$ , remains preserved if we require moreover $$\mathbb Q(\sqrt{D})$$ to contain an irrational with partial quotients bounded by A.