Values of non-homogeneous indefinite quadratic forms
Let \(f(X)\) be a real indefinite quadratic form in \(n\) variables of determinant \({\mathcal D}\neq 0\) and signature \(s\). It is well known that there exists a real number \(c>0\) depending only on \(n\) and \(s\) such that for any \(C\in \mathbb{R}^ n\) there exists \(X\in \mathbb{Z}\) such that \[ | f(X+C) |\leq (c | {\mathcal D}| )^{1/n}. \tag{1} \] The problem is to determine the infimum \(C_{n,s}\) of all such numbers \(c\). The first investigations gave the values of \(C_{n,s}\) only in some special cases. Then \textit{G. L. Watson} [Proc. Lond. Math. Soc., III. Ser. 12, 564-576 (1962; Zbl 0107.041)] gave a formula for \(C_{n,s}\) if \(n\geq 21\), \(s\) is arbitrary, and conjectured that the result is true for all \(n\geq 4\). Further investigations of the problem are finally completed in the present paper. So Watson's conjecture is established. If in the discussed problem the inequality (1) is changed to \(0< f(X+C)< (\Omega | {\mathcal D}|)^{1/n}\) with real \(\Omega> 0\), then the problem of determining the infimum \(\Omega_{n,s}\) of all such numbers \(\Omega\) arises. The authors prove that the values of \(\Omega_{n,s}\) for a fixed \(n\) are invariant under the signature modulo 8, what was conjectured by Bambah, Dumir and Hans-Gill. Now \(\Omega_{n,s}\) are determined for all meaningful pairs \(n\), \(s\) with the exception of \(\Omega_{6,-2}\) and \(\Omega_{5, -3}\). The proofs of both conjectures follow from the results of Margulis and Watson on rational forms.
scientific article; zbMATH DE number 584039