Abstract: The purpose of this talk is to present some harmonic analysis concepts and results used in establishing certain estimates on subharmonic functions. These estimates are basic ingredients in proving quantitative Birkhoff ergodic theorems for subharmonic sampling functions over base dynamics given by Diophantine rotations on one or higher dimensional tori. Such quantitative results are in turn used in the study of the Lyapunov exponents of quasi-periodic analytic co-cycles.