  We show that for every integer $d$, there is a constant $N(d)$ such that if
$K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non
amenable subgroup, then $F^{N(d)}$ contains two elements, which freely generate
a non abelian free subgroup. This improves the original statement of the Tits
alternative. It also implies a growth gap and a co-growth gap for non-amenable
linear groups, and has consequences about the girth and uniform expansion of
small sets in finite subgroups of $GL_d(\Bbb{F}_q)$ as well as other
diophantine properties of non-discrete subgroups of Lie groups.
