Determinant values on lattices
Abstract
We study the distribution of determinant values on lattices in Mn(ℝ) for n≥ 2. Let Λ<Mn(ℝ) be a lattice whose elements have algebraic entries. We prove that if det (Λ) is not contained in a scalar multiple of ℤ, then for every a<b, as T→ ∞, #{v∈Λ:|v| <T, a<det v<b, det v≠0} ∼ Cn/covol(Λ) (b−a)Tn(n−1) where |·| is the Frobenius norm and Cn>0 depends only on n. Under an additional hypothesis on the isotropic subspaces associated with Λ, which is automatic for n=2,3 and is satisfied when n≥ 4, by every lattice spanned by scalar multiples of the elementary matrices, we also obtain an asymptotic formula for the singular lattice points. Both conclusions extend to the broader class of Diophantine lattices under the corresponding hypotheses. For n=2, our theorem recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature (2,2); more generally, our results may be viewed as higher-degree analogues of that theorem.
Related
- Quantitative Oppenheim in signature (2,2) via determinant values
- The Cartan determinant conjecture for representation-finite Artin algebras
- Determinant characters and multiplicative solubility of finite skew braces
- A determinant formula for multiple harmonic sums and their linear independence
- Proof of Barry's Four Hankel Determinant Conjectures
BibTeX
@misc{kim2026determinant,
title = {{Determinant values on lattices}},
author = {Wooyeon Kim and Hee Oh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.18038},
eprint = {2607.18038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18038},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Determinant values on lattices
AU - Wooyeon Kim
AU - Hee Oh
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2607.18038
UR - https://arxiv.org/abs/2607.18038
AB - We study the distribution of determinant values on lattices in M_n(ℝ) for n≥ 2. Let Λ<M_n(ℝ) be a lattice whose elements have algebraic entries. We prove that if det (Λ) is not contained in a scalar multiple of ℤ, then for every a<b, as T→ ∞, #{v∈Λ:|v| <T, a<det v<b, det v≠0} ∼ C_n/covol(Λ) (b−a)T^n(n−1) where |·| is the Frobenius norm and C_n>0 depends only on n. Under an additional hypothesis on the isotropic subspaces associated with Λ, which is automatic for n=2,3 and is satisfied when n≥ 4, by every lattice spanned by scalar multiples of the elementary matrices, we also obtain an asymptotic formula for the singular lattice points. Both conclusions extend to the broader class of Diophantine lattices under the corresponding hypotheses. For n=2, our theorem recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature (2,2); more generally, our results may be viewed as higher-degree analogues of that theorem.
ER -