2026/07/20 by Wooyeon Kim, Hee Oh
#math.DS #math.NT
We give a new proof of the quantitative Oppenheim theorem in signature (2,2), originally proved by Eskin-Margulis-Mozes, by recasting the problem as one about determinant values on lattices in M2(\mathbb R). The determinant det\beginpmatrixx&y z&w\endpmatrix=xw-yz is a quadratic form of signature (2,2), and every real quadratic form of this signature is obtained from it by a real linear change of variables. For every Diophantine lattice Λ<M2(\mathbb R) that is not determinant-rational, for every a<b, we prove an asymptotic formula for #\v∈Λ:‖v‖<T, a<det v<b\. The main term has a nonsingular contribution proportional to (b-a)T2 and, when 0∈(a,b), a possible singular contribution from rational isotropic planes. The proof follows the modified-height and avoidance strategy, but in the 2×2 case the representation theory and sublevel estimates are elementary, and the rational isotropic planes form a finite collection for a non-determinant-rational lattice.