  We prove that cocompact arithmetic lattices in a simple Lie group are
uniformly discrete if and only if the Salem numbers are uniformly bounded away
from $1$. We also prove an analogous result for semisimple Lie groups. Finally,
we shed some light on the structure of the bottom of the length spectrum of an
arithmetic orbifold $\Gamma\backslash X$ by showing the existence of a positive
constant $\delta(X)>0$ such that squares of lengths of closed geodesics shorter
than $\delta$ must be pairwise linearly dependent over $\mathbb Q$.
