Values of quadratic forms at primitive integral points
The authors continue \textit{G. Margulis}' investigation of the set of values of real indefinite quadratic forms with the help of ergodic methods [Proc. Symp. in Honor of A. Selberg, Oslo 1987, 377--398 (1989; Zbl 0675.10010)]). Let \(B\) be such a nondegenerate form in at least 3 variables that is not proportional to a rational form and \(T\) a binary form represented by \(T\) over the reals. Then it is proved that there are binary forms arbitrary close to \(T\) that are represented by \(B\) with primitive integral representing vectors. (It would be interesting to know whether the representing pair of integral vectors can be chosen such that it spans a direct summand of \(\mathbb Z^n)\). The result is shown to depend on the fact that all orbits in \(\mathrm{SL}(3,\mathbb R)/\mathrm{SL}(3,\mathbb Z)\) of the connected component of 1 of the special orthogonal group of B are either closed or dense. The proof of this fact (and results on flows on homogeneous spaces of Lie groups necessary for it) occupy the main part of the paper.
values of real indefinite quadratic forms
flows on homogeneous spaces of Lie groups
scientific article; zbMATH DE number 4116840
scientific article