Bounded generation in linear groups and exponential parameterizations

In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine $S$-unit theory to problems of “bounded generation” in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on “anisotropic groups” (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are “sparse”. Actually, this holds for all sets obtained by exponential parameterizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.