Purely exponential parametrizations and their group-theoretic applications

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.)