2026/07/20 by Wooyeon Kim, Hee Oh · 1 citation
#math.DS #math.NT
We study the distribution of determinant values on lattices in Md(\mathbb R) for d≥ 2. Let Λ<Md(\mathbb R) be a lattice whose elements all have algebraic entries. We prove that if det (Λ) is not contained in a scalar multiple of \mathbb Z, then for every a<b, #\v∈Λ:‖v‖ <T, a<det v<b, det v≠0\ ∼ (Cd)/(covol(Λ)) (b-a)Td(d-1) as T→ ∞, where ‖⋅‖ is the Frobenius norm and Cd>0 depends only on d. For such a lattice, under an isotropic noncoincidence hypothesis, automatic for d=2,3 and satisfied for all diagonal lattices when d≥ 4, we also obtain an asymptotic formula for the determinant-zero lattice points. The same conclusions hold for the broader class of Diophantine lattices, under the corresponding hypotheses. For d=2, our result recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature (2,2).