Oppenheim conjecture

In mathematics, the Oppenheim conjecture is now proven conjecture about the values ​​of quadratic forms and the classic example of the application ergodentheoretischer methods in number theory.

Statement

Be and

An indefinite quadratic form in n variables which is not a multiple of a form with rational coefficients.

Then there is for each a with

As a corollary we obtain that a dense subset of is.

Example: For each there are integers with

History

The assumption in this form was erected in 1953 ( a weaker predecessor version in 1929 ) of Oppenheim and proved for Birch, Davenport and Ridout. The general case can be traced back to the case and this has been reworded by Raghunathan in the following conjecture on the left- action of on the quotient space:

This conjecture was proved in 1987 by Margulis. A more general version of Raghunathan 's conjecture is the current set of Ratner.

• Theory of dynamical systems
• Number Theory