Formes quadratiques indéfinies et flots unipotents sur les espaces homogènes. (Indefinite quadratic forms and unipotent flows on homogeneous spaces).

*(English)*Zbl 0624.10011By a theorem of A. Meyer (1884), we know that any indefinite quadratic form with rational coefficients in more than 4 variables represents 0 for integral values of variables not all zero. It has been a longstanding conjecture, on the other hand, that any indefinite quadratic form in \(n\geq 5\) variables with real coefficients, which is not a scalar multiple of a form with rational coefficients, assumes values arbitrarily close to 0, for integral values of the variables.

There have been many contributions towards the confirmation of this remarkable conjecture, starting from that of S. D. Chowla (1934), next from H. Heilbronn and H. Davenport (1946), and then the papers of B. J. Birch, H. Davenport and D. Ridout which established the validity of the conjecture for \(n\geq 21\). The present magnificent paper establishes the validity of “Davenport’s conjecture” on values of “irrational” indefinite quadratic forms even for \(n\geq 3\) variables, using results on closed subsets of homogeneous spaces invariant under unipotent subgroups.

Using K. Mahler’s well-known compactness criterion and a famous density theorem due to A. Borel, the proof for the conjecture is reduced to the following interesting theorem: Let \(G=SL(3, {\mathbb{R}})\), \(\Gamma =SL(3, {\mathbb{Z}})\), \(G_ x:=\{g\in G|\) \(gx=x\}\) for x in G/\(\Gamma\) and H, the orthogonal group of the ternary quadratic form \(2x_ 1x_ 3-x^ 2_ 2\). If the orbit Hx is relatively compact in the homogeneous space G/\(\Gamma\), then \(H/H\cap G_ x\) is compact.

It was pointed out by M. S. Raghunathan that “Davenport’s conjecture” would follow from a related conjecture on the structure of orbits of unipotent subgroups U, in the case of semisimple Lie groups. The latter conjecture was confirmed in the case of horospherical subgroups U, by recent work of S. G. Dani; the analogue of the same conjecture in the case of the above H (in place of U) now stands verified for relatively compact orbits.

There have been many contributions towards the confirmation of this remarkable conjecture, starting from that of S. D. Chowla (1934), next from H. Heilbronn and H. Davenport (1946), and then the papers of B. J. Birch, H. Davenport and D. Ridout which established the validity of the conjecture for \(n\geq 21\). The present magnificent paper establishes the validity of “Davenport’s conjecture” on values of “irrational” indefinite quadratic forms even for \(n\geq 3\) variables, using results on closed subsets of homogeneous spaces invariant under unipotent subgroups.

Using K. Mahler’s well-known compactness criterion and a famous density theorem due to A. Borel, the proof for the conjecture is reduced to the following interesting theorem: Let \(G=SL(3, {\mathbb{R}})\), \(\Gamma =SL(3, {\mathbb{Z}})\), \(G_ x:=\{g\in G|\) \(gx=x\}\) for x in G/\(\Gamma\) and H, the orthogonal group of the ternary quadratic form \(2x_ 1x_ 3-x^ 2_ 2\). If the orbit Hx is relatively compact in the homogeneous space G/\(\Gamma\), then \(H/H\cap G_ x\) is compact.

It was pointed out by M. S. Raghunathan that “Davenport’s conjecture” would follow from a related conjecture on the structure of orbits of unipotent subgroups U, in the case of semisimple Lie groups. The latter conjecture was confirmed in the case of horospherical subgroups U, by recent work of S. G. Dani; the analogue of the same conjecture in the case of the above H (in place of U) now stands verified for relatively compact orbits.

Reviewer: S.Raghavan

##### MSC:

11D75 | Diophantine inequalities |

22E40 | Discrete subgroups of Lie groups |

22E25 | Nilpotent and solvable Lie groups |