Oppenheim conjecture

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

statement

Be and ${\displaystyle n\geq 3}$

${\displaystyle Q:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$

an indefinite quadratic form in variables that is not a multiple of a form with rational coefficients. ${\displaystyle n}$

Then there is one with for each ${\displaystyle \epsilon >0}$ ${\displaystyle x\in \mathbb {Z} ^{n}}$

${\displaystyle 0<Q(x)<\epsilon }$.

As a corollary we get that is a dense subset of . ${\displaystyle Q(\mathbb {Z} ^{n})\subset \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

Example : For each there are whole numbers with ${\displaystyle \epsilon >0}$ ${\displaystyle a,b,c}$

${\displaystyle \mid a^{2}+b^{2}-{\sqrt {2}}c^{2}\mid <\epsilon }$.

history

The conjecture in this form was set up in 1953 (a weaker predecessor version as early as 1929) by Alexander Oppenheim and proven for by Bryan Birch , Harold Davenport and D. Ridout. The general case can be traced back to the case and this was reformulated by MS Raghunathan into the following conjecture about the left effect of on the quotient space : ${\displaystyle n\geq 21}$ ${\displaystyle n=3}$ ${\displaystyle SO(2,1)}$ ${\displaystyle SL(3,\mathbb {R} )/SL(3,\mathbb {Z} )}$

Each constrained orbit at is compact. ${\displaystyle SO(2,1)}$ ${\displaystyle SL(3,\mathbb {R} )/SL(3,\mathbb {Z} )}$

This conjecture was proven in 1987 by Grigori Margulis . A more general version of the Raghunathan conjecture is today's Ratner Theorem .