Oppenheim conjecture

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 later gave a general picture of such orbit closures, as predicted by Raghunathan's conjecture. 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 $0$.

Going deeper

For a student

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 $a < Q(x) < b$.