---
tag: 4S50G4PS26
kind: statement
role: theorem
name: Oppenheim conjecture
msc: ["11E20", "37A17", "22E40"]
uses: [0RBQS1JJCR, 01RBSSGMY5, 0BSJSX7ME2]
about: [0RBQS1JJCR]
sources:
  - url: https://en.wikipedia.org/wiki/Oppenheim_conjecture
same_as:
  wikidata: Q7098547
  wikipedia: Oppenheim conjecture
---

## Statement

Let $n \ge 3$ and let $Q$ be a real [[quadratic form|0RBQS1JJCR]] in $n$ variables.
Assume that $Q$ is nondegenerate, that $Q$ is [[indefinite|01RBSSGMY5]], and that $Q$ is not a real multiple of a form with rational coefficients.
Then the set of values $Q(\Z^n) = \set{Q(x) : x \in \Z^n}$ is [[dense|0BSJSX7ME2]] in $\R$.
In particular, for every $\varepsilon > 0$ there is $x \in \Z^n \setminus \set{0}$ with $0 < \abs{Q(x)} < \varepsilon$.

## Remarks

Margulis proved the statement in 1987, so the name "conjecture" is historical.
Following an idea of Raghunathan, it reduces to the case $n = 3$ and to the orbits of the group $\mathrm{SO}(Q)$ on the space $\mathrm{SL}_3(\R)/\mathrm{SL}_3(\Z)$ of unimodular lattices in $\R^3$.
The heart of the proof is Margulis's theorem that a relatively compact orbit of $\mathrm{SO}(2,1)$ in this space is compact, while the irrationality of $Q$ rules out a compact orbit.
[[Ratner's theorems|3MXR03096V]] later gave a general picture of such orbit closures, as predicted by [[Raghunathan's conjecture|ME4ZSRAT5Z]].
An example: $Q(x, y, z) = x^2 + y^2 - \sqrt{2}\, z^2$ satisfies the hypotheses, so its values at integer points are dense in $\R$.
Each hypothesis is needed.
If $Q$ is definite, the values on $\Z^n$ form a discrete set.
If $Q$ is a multiple $c Q_0$ of a rational form $Q_0$ with denominators dividing $d$, its values lie in the discrete set $(c/d)\Z$.
For $n = 2$ the form $x^2 - \varphi^2 y^2 = (x - \varphi y)(x + \varphi y)$, with $\varphi$ the golden ratio, is irrational and indefinite.
But $\abs{x - \varphi y} \ge c/\abs{y}$ for a constant $c > 0$ and all integers $x$ and $y \ne 0$.
This shows that the nonzero values of the form at integer points stay away from $0$.

## Explanations

### L1 human

A quadratic form is a polynomial like $x^2 + y^2 - \sqrt{2}\, z^2$.
Feed it only whole numbers and look at the outputs.
The theorem says that, with at least three variables, mixed signs and an irrational coefficient that cannot be scaled away, the outputs come arbitrarily close to every real number.
This is a statement about integers that was proved with the dynamics of group actions.

### L2 human

The theorem was a turning point for the use of homogeneous dynamics in number theory.
The key move is to replace integer points by the lattice $\Z^3$.
Then one moves this lattice by elements of the orthogonal group of $Q$, which do not change the values of $Q$.
Then density of values becomes a statement about the closure of an orbit of $\mathrm{SO}(2,1)$ in the space of lattices.
This viewpoint led to Ratner's classification of orbit closures for unipotent flows and to quantitative versions counting integer points with $a < Q(x) < b$.