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Quadratic forms of signature $$(2,2)$$ and eigenvalue spacings on rectangular 2-tori. (English) Zbl 1075.11049
A quantitative version of the Oppenheim conjecture proved by Margulis states that for a nondegenerate indefinite quadratic form $$Q$$ in $$n$$ variables there exists a constant $$\lambda_{Q,\Omega}$$ such that for any interval $$(a b)$$ as $$T\to\infty$$ $$\text{Vol}\{x\in\mathbb{R}^n: x\in T\Omega$$ and $$a\leq Q(s)\leq b\}\sim\lambda_{Q,\Omega(b- a)T^{n-2}}$$, where $$\Omega= \{v\in\mathbb{R}^n\mid\| v\|<\rho(v/\| v\|)\}$$ and $$\rho$$ is a continuous positive function on the sphere $$\{v\in\mathbb{R}^n\mid\| v\|=1\}$$. Eskin, Margulis and Mozes have shown that $$N_{Q,\Omega}(a,b,T)\sim \lambda_{Q,\Omega}(b- a)T^{n-2}$$ where $$Q$$ is an indefinite quadratic form (not proportional to a rational form) of signature $$(p, q)$$ with $$p\geq 3$$, $$q\geq 1$$, $$n= p+ q$$ and $$N_{Q,\Omega}(a, b, T)$$ denotes the cardinality of the set $$\{x\in\mathbb{Z}^n:x\in T\Omega$$ and $$a< Q(x)< b\}$$. If the signature of $$Q$$ is $$(2,1)$$ or $$(2,2)$$ then the above result fails. Whenever a form of signature $$(2,2)$$ has a rational isotropic subspace $$L$$ then $$L\cap T\Omega$$ contains on the order of $$T^2$$ integral points $$x$$ for which $$Q(x)= 0$$, hence $$N_{Q,\Omega}(-\varepsilon, \varepsilon, T)\geq cT^2$$, independently of the choice of $$\varepsilon$$.
Thus to obtain an asymptotic formula in the signature $$(2,2)$$ case, we must exclude the contribution of the rational isotropic subspaces. The main result of this paper is as follows: Let $$Q$$ be an indefinite quadratic form of signature $$(2,2)$$ which is not extremely well approximable by split forms then for any interval $$(a, b)$$ as $$T\to\infty$$, $$\widetilde N_{Q,\Omega}(a, b, T)\sim\lambda_{Q,\Omega}(b- a)T^2$$ where $$\widetilde N_{Q,\Omega}$$ counts the points not contained in isotropic subspaces. It turns out that points belonging to a wider class of subspaces have to be treated separately. In order to estimate $$N_{Q,\Omega}$$ a transition to considering certain integrals on the space of unimodular lattices in $$\mathbb{R}^n$$ is made. This transition is based on the transitivity of the action of the orthogonal group $$\text{SO}(Q)$$ on the level sets of the quadratic form $$Q$$. One also needs to have an estimate of the contribution of elements of lattices lying at the cusps of $$\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{Z})$$.

##### MSC:
 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 22E40 Discrete subgroups of Lie groups
##### Keywords:
quadratic form; isotropic; orthogonal group; unimodular
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