Mathematical Research Letters
Volume 22 (2015)
On values of binary quadratic forms at integer points
Pages: 1023 – 1045
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $\lvert Q(x, y) \rvert \lt \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincaré half-plane.