On values of binary quadratic forms at integer points

Manoj Choudhuri, S. G. Dani

Published 2014-04-21, updated 2014-09-26Version 2

We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < \epsilon$, 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 Poincare half-plane.