  In this paper, we consider the polynomial and exponential convergence rate of
weighted Birkhoff averages of irrational rotations on tori. It is shown that
these can be achieved for finite and infinite dimensional tori which correspond
to the quasiperiodic and almost periodic dynamical systems respectively, under
certain balance between the nonresonant condition and the decay rate of the
Fourier coefficients. Diophantine rotations with finite and infinite dimensions
are provided as examples. For the first time, we prove the universality of
exponential convergence and arbitrary polynomial convergence in the
quasiperiodic case and almost periodic case under analyticity respectively.
