A few months ago, I attended the very interesting conference “Conférence Internationale Géométrie Ergodique” held at the Math. Department of Orsay (Univ. Paris 11) from 23th to 27th May. My general feeling was that the conference was very exciting and the lecturers did a great effort to make their expositions as clear as possible. In particular, I decided to transcript my notes for the lectures of Elon Lindenstrauss, Manfred Einsiedler, Dmitry Kleinbock and Yves Guivarch.
So, let’s get started with Elon Lindenstrauss’ talk (based on joint work with Gregory Margulis). The title of Lindenstrauss’ talk was:
1. An effective proof of Oppenheim conjecture
Conjecture (Oppenheim). Let be an indefinite quadratic form in variables such that is not proportional to an integral form (i.e., a form with integer coefficients). Then,
In this form, the conjecture is due to Davenport. The original conjecture by Oppenheim was for variables, inspired by
Theorem 1 (Meyer) If is an integral indefinite quadratic form in , then represents non-trivially, i.e., for some .
Remark 1 doesn’t represent non-trivially (check everything mod 3). So, for this , a quadratic form in 4 variables, .
Oppenheim showed that his conjecture is true for , . His proof is based on two facts:
- any odd number mod can be written as ;
- mod is dense.
Davenport and his coauthors proved (around 1946) Oppenheim’s conjecture for:
- , (i.e., diagonal quadratic forms in variables);
- general quadratic forms in variables.
G. Margulis proved Oppenheim’s conjecture in mid 80’s using homogenous dynamics: in fact, Margulis proved that if is a quadratic indefinite form not proportional to integral in variables, then
After that, S. Dani and G. Margulis extended this result to show, e.g., that under the hypothesis of Oppenheim’s conjecture, .
The restriction is necessary:
Here, can be replaced by any number which is the square of a badly approximable number, e.g., .
However, this is not the case of Margulis dynamical proof.
Margulis’ proof goes like this. Firstly, one noticed that it suffices to proved Oppenheim’s conjecture for quadratic form in variables (i.e., ternary quadratic forms). Indeed, it is not hard to check that if we start with an indefinite quadratic form in variables which is not proportional to an integral form, then we can find a “generic” and “irrational” hyperplane such that the restriction of to this hyperplane is a quadratic form in variables which is both indefinite and not proportional to integer. Thus, if we can prove the conjecture in variables, we will be able to conclude it in variables.
Secondly, for the case of 3 variables, Margulis proved the conjecture by proving the following dynamical result. Let be the group of matrices preserving .
Here, we say that is periodic if the stabilizer has finite covolume in .
Remark 3 Since this definition of periodicity asks only for finite covolume (and not cocompactness), the conditions “periodic” and “unbounded” are not mutually exclusive.
As it was already mentioned, S. Dani and G. Margulis improved this theorem by showing that:
To see why the dynamical results from Theorems 2 and 3 have something to do with Oppenheim conjecture (and Corollary 4 above), we observe that any quadratic indefinite form (of signature ) can be written as with . In this notation,
An easy argument gives that
So, by Dani-Margulis theorem, is dense whenever is as in Oppenheim conjecture. Now, we note that, somewhere in (the space of unimodular lattices), there is a lattice with and primitive, so that there is such that contains a primitive vector very close to , and hence
Closing the “historical introduction” part of the talk, Lindenstrauss remarked that Dani-Margulis theorem is a special case of Raghunathan’s conjecture which in particular classifies orbit closures in whenever is generated by unipotent one-parameter subgroups. Furthermore, Raghunathan’s conjecture was proved in full generality by Marina Ratner.
Remark 4 Of course, this is a very long history and due to the usual space-time limitations, Lindenstrauss was obliged to stop here the introductory considerations. For the curious reader wishing to learn more about the topics mentioned above, I strongly recommend reading Terence Tao’s posts on this subject, specially these ones here.
Anyway, as it was more or less hinted in Remark 2 above, it is not very easy to convert Margulis’ proof directly into an effective/quantitative statement. However, in their way to extend Margulis’ result, S. Dani and G. Margulis observed that there is a shortcut giving some hope for quantitative theorems:
Remark 5 (Dani-Margulis’ shortcut) To establish Corollary 4, it is enough to show that, if is not periodic, then there exists such that , where
This “shortcut” allowed E. Lindenstrauss and G. Margulis to recently prove the following series of theorems:
Disclaimer: While I did my best to get correct quantifiers in the statements below, it could be that I introduced errors at the moment of taking notes. In particular, any mistakes in the statements below are, of course, my fault.
Theorem 5 Let be an indefinite ternary quadratic form such that . Assume that the coefficients of are algebraic but is not proportional to an integral form. Then, for any , there exists primitive, such that
Here, is an uniform constant and depends only on the “algebraic degree” (height) of the coefficients of .
Theorem 6 For every , there exists a constant with the following property. Let be indefinite quadratic form in variables with and all coefficients of are in absolute value. Assume that, for , there is no integral with such that
Then, for all , there is a primitive such that and
Here, , and are uniform constants (depending only on ).
Remark 6 For sake of comparison between these two theorems, we notice that their common “feature” is the presence of a “Diophantine condition”: in the first theorem, this corresponds to the assumption that the coefficients of are algebraic (but is not a multiple of an integral form), while in the second theorem this is expressed by imposing that is badly approximable by integral forms (in a precise quantitative way).
For the statement of the next theorem, we’ll need the following conventions:
- , i.e., the subset of consisting of all lattices with no -short vectors;
- for any , , .
Finally, E. Lindenstrauss announced that G. Margulis and him were also capable of producing a quantitative version of the “ is periodic or dense” dichotomy (Dani-Margulis Theorem 3) saying that, if we can avoid certain periodic orbits nearby (more or less like in Theorem 7 above), then the orbit must be dense. However, since E. Lindenstrauss started running out of time, he could neither give a precise statement for this last claim/theorem nor give sketches of proofs of any of the 3 theorems stated above.