Inhomogeneous minima of a class of quaternary quadratic forms of type \((3,1)\)
Let \(Q(x_1,\dots,x_n)\) be a real indefinite quadratic form in \(n\) variables of determinant \(D\neq 0\) and of type \((r,n-r)\). Let \(\Gamma_{r,n-r}\) denote the infimum of all numbers \(\Gamma>0\) such that for any real numbers \(c_1,c_2,\dots, c_n\) there exist \((x_1,\dots,x_n)\equiv(c_1,\dots,c_n)\pmod 1\) satisfying \[ 0<Q(x_1,\dots,x_n)\leq\bigl(\Gamma|D|\bigr)^{1/n}\tag{1} \] The values of \(\Gamma_{r,n-r}\) are known for all \(n\) and \(r\) except for \(\Gamma_{1,4}\); for reference see \textit{V. C. Dumir} and \textit{R. Sehmi} [Proc. Indian Acad. Sci., Math. Sci. 104, 557-579 (1994; Zbl 0808.11044)]. It was shown by \textit{H. Davenport} that \(\Gamma_{1,1}\) is not isolated. For \(n\geq 3\), L. Vulakh has shown that \(\Gamma_{r,n-r}\) is isolated for rational forms, whereas for irrational forms, it follows from results of G.~L. Watson and G. Margulis that (1) is solvable for arbitrary small values of \(\Gamma\). For \(n\geq 3\) let \(\Gamma^{(k)}_{r,n-r}\) denote the \(k\)th successive inhomogeneous minimum. The values of \(\Gamma^{(2)}_{2,1}\), \(\Gamma^{(2)}_{1,2}\), \(\Gamma^{(2)}_{r+2,r}\), \(\Gamma^{(2)}_{2,3}\), \(\Gamma^{(3)}_{2,3}\), \(\Gamma^{(4)}_{2,3}\) and \(\Gamma^{(2)}_{r+1,r}\) have been obtained by various authors; for reference see \textit{M. Raka} and \textit{U. Rani} [Sitzungsber., Abt. II, Akad. Wiss., Math.-Naturwiss. Kl. 203, 175-197 (1994; Zbl 0838.11049)]. In this paper the first four minima namely 16/3, 4, 16/5 and 81/32 for positive values of zero quaternary quadratic forms of type (3,1) have been obtained. All the critical forms are also determined.
scientific article; zbMATH DE number 1281616