We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if Salem numbers are uniformly bounded away from
1
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
Γ
∖
X
\Gamma \backslash X
by showing the existence of a positive constant
δ
(
X
)
>
0
\delta (X)>0
such that squares of lengths of closed geodesics shorter than
δ
\delta
must be pairwise linearly dependent over
Q
\mathbb {Q}
.
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