New bounds in reduction theory of indefinite ternary integral quadratic forms (Q1701013)

The authors obtain some new and improved estimates for reduction of indefinite ternary integral quadratic forms using dynamics on homogeneous spaces. A main result is the following theorem: There is a constant \(C>0\) such that for any indefinite ternary integral quadratic form \(Q_1\) and \(\xi\in(-1,1)\), there exists a quadratic form \(Q= (a_{ij})\) equivalent to \(Q_1\) and satisfying \[ \Biggl|{a_{11}\over(\text{det}\,Q_1)^{{1\over 3}}}-\xi\Biggr|<C\cdot\text{vol}(Q_1)^{-{1\over 80}}, \] \[ \operatorname{ht}(Q)< C\cdot|\text{det\,}Q_1|^{{1\over 3}}, \] where \(\operatorname{ht}(Q)= \max(|a_{ij}|)\) is the height and \(\text{vol}(Q_1)\) is a certain co-volume with respect to a certain Haar measure.