Equidistribution of values of linear forms on quadratic surfaces
In this paper, the author investigates the problem of distribution of the sets of values of a linear map at integer points on a quadratic surface. This problem is motivated by a rich literature on the Oppenheim conjecture which states that if \(d \geq 3\) and \(Q\) is a real nondegenerate indefinite quadratic form on \(\mathbb{R}^d\), which is not a multiple of a form with rational coefficients, then for any \(\varepsilon>0\) there exists a non-zero vector \(x\) with integer components such that \(|Q(x)|<\varepsilon\). This conjecture was proved by \textit{S. G. Dani} and \textit{G. A. Margulis} [Math. Ann. 286, No. 1--3, 101--128 (1990; Zbl 0679.22007)]. The main result of this paper is as follows:NEWLINENEWLINE Theorem: Suppose \(Q\) is a quadratic form on \(\mathbb{R}^d\) such that \(Q\) is nondegenerate and indefinite with rational coefficients. Let \(M = (L_1,\cdots,L_s)\;:\;\mathbb{R}^d \longrightarrow \mathbb{R}^s\), be a linear map such that:NEWLINENEWLINE(1) The following relations hold: \(d > 2s\) and \({\mathrm{rank}}(Q|_{\ker(M)}) = d -s\).NEWLINENEWLINE(2) The quadratic form \(Q|_{\ker(M)}\) has signature \((r_1 ,r_2 )\), where \(r_1 \geq 3\) and \(r_2 \geq 1\).NEWLINENEWLINE(3) For all \(\alpha=(\alpha_1,\cdots,\alpha_s) \in \mathbb{R}^s\setminus\{0\}\), \(\alpha_1 L_1 +\cdots+\alpha_s L_s\) is nonrational.NEWLINENEWLINELet \(a \in \mathbb{Q}\) be such that the set \(\big\{v \in \mathbb{Z}^d\;: \;Q(v) = a~\big\}\) is nonempty. Then, there exists \(C_0 > 0\) such that, for every \(\varepsilon > 0\) and all compact \(K \subset \mathbb{R}^s\) with piecewise smooth boundary, there exists a \(T_0 > 0\) such that, for all \(T > T_0\) , NEWLINE\[NEWLINE(1-\varepsilon)C_0 {\operatorname{Vol}}(K) \leq \frac{|\big\{v \in \mathbb{Z}^d\;: \;Q(v) = a, M(v) \in K, \|v\| \leq T\big\}|}{T^{d-s-2}} \leq (1+\varepsilon)C_0 {\operatorname{Vol}}(K),NEWLINE\]NEWLINE where \({\operatorname{Vol}}(K)\) is the \(s\)-dimensional Lebesgue measure of \(K\).NEWLINENEWLINE The methods of the proof are based on those developed by \textit{A. Eskin} et al. [Ann. Math. (2) 147, No. 1, 93--141 (1998; Zbl 0906.11035)].
scientific article; zbMATH DE number 6331626