Bounded Generation in Linear Groups and Diophantine Approximation

A group is said to be boundedly generated if it is decomposable as a product of finitely many cyclic groups. In this survey, we explore the link between bounded generation in matrix groups, linear recurrence sequences, exponential Diophantine equations and Diophantine approximation. In particular, we present recent results obtained by the author in collaboration with J. Demeio, A. Rapinchuk, J. Ren and U. Zannier, which show that bounded generation cannot hold for a large class of groups of matrices.