Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms
Let \(Q\) be a non-degenerate indefinite quadratic form on \(\mathbb R^n\). Let \(\xi\in\mathbb R^n\) be a vector, and define the (inhomogeneous) quadratic form \(Q_\xi\) by \(Q_\xi= Q(x+\xi)\) for all \(x\in\mathbb R^n\). Let \(v\) be a continuous function on the sphere \(\{v\in\mathbb R^n:\| v\|= 1\}\). Define \(\Omega= \{v\in\mathbb R^n:\| v\|< \nu(v/\| v\|\}\), and let \(T\Omega\) be the dilate of \(\Omega\) by \(T\). Let \(N_{Q,\xi,\Omega}= \#\{x\in\mathbb Z^n: x\in T\Omega\) and \(a< Q_\xi(x)< b\}\). If \(\xi= 0\), we let \(N_{Q,\xi,\Omega}(a,b,T)= N_{Q,\Omega}(a,b, T)\). It is easy to see that there exists a constant \(\lambda_{Q,\Omega}\) such that \(\text{Vol}(\{x\in\mathbb R^n: x\in T\Omega\) and \(a< Q_\xi(x)< b\})\sim\lambda_{Q,\xi}(b- a)T^{n-2}\). \textit{A. Eskin}, the first author and \textit{S. Mozes} [Electron. Res. Announc. Am. Math. Soc. 1, No. 3, 124--130 (1995; Zbl 0859.11043); Ann. Math. (2) 147, No. 1, 93--141 (1998; Zbl 0906.11035); Ann. Math. (2) 161, No. 2, 679--725 (2005; Zbl 1075.11049)] proved that if \(Q\) be a quadratic form not proportional to a rational form and of signature \((p,q)\) with \(p\geq 3\) and \(q\geq 1\), then for any interval \((a,b)\), we have \[ N_{Q,\Omega}(a,b,T)\sim \lambda_{Q,\xi}(b- a)T^{n-2} \] as \(T\to\infty\), where \(n= p+q\). If \(Q\) is an irrational quadratic form of signature \((2,2)\) which is not extremely well approximable by split forms (EWAS) then \[ \widetilde N_{Q,\Omega}(a, b, T)\sim\lambda_{Q,\xi}(b- a)T^2 \] as \(T\to\infty\), where \(\widetilde N_{Q, \xi, \Omega}(a,b,T)= \#\{x\in\mathbb Z^n: x\) is not a null subspace of \(Q_\xi\), \(x\in T\Omega\) and \(a< Q_\xi(x)< b\}\). This paper extends the above results to inhomogeneous quadratic forms as follows: 1. Let \(Q_\xi\) be an inhomogeneous quadratic form of signature \((2,2)\). Assume that \(Q_\xi\) is Diophantine. Then for any interval \((a,b)\), we have \[ \widetilde N_{Q,\xi,\Omega}(a,b,T)\sim \lambda_{Q,\Omega}(b-a)T^2 \] as \(T\to\infty\). 2. Let \(Q_\xi\) be an in quadratic form of signature \((2,1)\). If the homogeneous part \(Q\) is a split rational form and if the vector \(\xi\) is Diophantine, then for any interval \((a,b)\), we have \[ \widetilde N_{Q,\xi,\Omega}(a,b,T)\sim \lambda_{Q,\Omega}(b-a)T \] as \(T\to\infty\). The proof of the first result requires the additional case when the homogeneous part \(Q\) is EWAS as compared to ideas used by \textit{A. Eskin}, the first author and \textit{S. Mozes} [loc. cit.]. An especially important case is when \(Q\) is a split rational form. In this case we need to study the contribution coming from the null subspaces to the counting function \(\widetilde N\). In case of forms with signature \((2,1)\) short vectors and thin planes interfere with each other. This prevents from successfully using the ideas of \textit{A. Eskin}, the first author and \textit{S. Mozes} [loc. cit.]. The absence of such an asymptotic in the homogeneous case with explicit Diophantine condition is responsible for the extra assumption in the second result above that the homogeneous part \(Q\) is a split rational form. The Diophantine condition is necessary in both results. An application to eigenvalue spacing on flat 2-tori with Aharonov-Bohm flux is also included in this paper.
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