A proof of Margulis' theorem on values of quadratic forms, independent of the axiom of choice (Q1332390)

Let \(Q\) be a nondegenerate indefinite quadratic form on \(\mathbb{R}^ n\), \(n\geq 3\), which is not a scalar multiple of a rational form. \textit{G. Margulis} [C. R. Acad. Sci., Paris, Sér. I 304, 249-253 (1987; Zbl 0624.10011)] proved that the set \(Q( \mathbb{Z}^ n)\) of values of \(Q\) on the set of integral \(n\)-tuples is a dense subset of \(\mathbb{R}\), thereby proving the longstanding conjecture of A. Oppenheim. \textit{S. G. Dani} and \textit{G. Margulis} [Invent. Math. 98, 405-424 (1989; Zbl 0682.22008)] strengthened the result by proving the density of the set of values of \(Q\) on the set of primitive integral \(n\)-tuples. In a subsequent paper, \textit{S. G. Dani} and \textit{G. Margulis} [Enseign. Math. 36, 143-174 (1990)] gave an elementary proof of this result based only on standard arguments in topological groups and linear algebra. \textit{G. Margulis} [Isr. Math. Conf. Proc. 3, 127-150 (1990; Zbl 0718.11027)] and \textit{J.-C. Sikorav} [Prog. Math. 81, 307-315 (1990; Zbl 0704.11009)] proved some similar but weaker results. All these proofs, somehow used the axiom of choice. In this paper a variation of the proof of Dani and Margulis (loc. cit.) is given which does not depend on the axiom of choice. The proof is based on orbits of certain actions on the homogeneous space \(\text{SL} (3,\mathbb{R})/ \text{SL} (3,\mathbb{Z})\) and uses some results from the paper of Dani and Margulis. The present result perhaps can also be deduced from \textit{M. Ratner}'s proof [Duke Math. J. 63, 235-280 (1991; Zbl 0733.22007)] of Raghunathan's conjecture.