AI 中文总结
研究了晶格上行列式值的分布，证明了在特定条件下行列式值的渐进行式，并恢复了关于二次型的定理。
AI 中文摘要
我们研究了在$\operatorname{M}_d(\mathbb R)$中晶格上行列式值的分布，其中$d\ge 2$。令$\Lambda<\operatorname{M}_d(\mathbb R)$是一个所有元素都有代数项的晶格。我们证明，如果$\det (\Lambda)$不包含在标量倍数的$\mathbb Z$中，那么对于每一个$a<b$，$$ \#\{v\in\Lambda:\|v\| <T,\ a<\operatorname{det} v<b,\ \operatorname{det} v\ne0\} \sim \frac{C_d}{\operatorname{covol}(\Lambda)} (b-a)T^{d(d-1)} $$当$T\to \infty$时成立，其中$\|\cdot\|$是Frobenius范数，$C_d>0$仅依赖于$d$。对于这样的晶格，在等方非重合假设下，自动适用于$d=2,3$，并且在$d\ge 4$时适用于所有对角晶格，我们还获得了行列式为零的晶格点的渐进行式。对于$d=2$，我们的结果恢复了Eskin-Margulis-Mozes关于二次型签名$(2,2)$的定量Oppenheim问题的定理。
英文摘要
We study the distribution of determinant values on lattices in $\operatorname{M}_n(\mathbb R)$ for $n\ge 2$. Let $Λ<\operatorname{M}_n(\mathbb R)$ be a lattice whose elements have algebraic entries. We prove that if $\det (Λ)$ is not contained in a scalar multiple of $\mathbb Z$, then for every $a<b$, as $T\to \infty$, $$ \#\{v\inΛ:\|v\| <T,\ a<\operatorname{det} v<b,\ \operatorname{det} v\ne0\} \sim \frac{C_n}{\operatorname{covol}(Λ)} (b-a)T^{n(n-1)} $$ where $\|\cdot\|$ 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 $n\geq 4$, 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.
Comments137 pages
