  This paper is mainly motivated by the analysis of the so-called Bounded
Generation property (BG) of linear groups (in characteristic $0$), which is
known to admit far-reaching group-theoretic implications.
  We achieve complete answers to certain longstanding open questions about
Bounded Generation (sharpening considerably some earlier results). For
instance, we prove that linear groups boundedly generated by semi-simple
elements are necessarily virtually abelian. This is obtained as a corollary of
sparseness of subsets which are likewise generated.
  In the paper in fact we go further, framing (BG) in the more general context
of (Purely) Exponential Parametrizations (PEP) for subsets of affine spaces, a
concept which unifies different issues. Using deep tools from Diophantine
Geometry (including the Subspace Theorem), we systematically develop a theory
showing in particular that for a (PEP) set over a number field, the asymptotic
distribution of its points of Height at most $T$ is always $\sim c(\log T)^r$,
with certain constants $c>0$ and $r\in \mathbb{Z}_{\geq 0}$. (This shape fits
with a well-known viewpoint first put forward by Manin.)
