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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
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