Precise asymptotics on the Birkhoff sums for dynamical systems

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These are parallel to that arithmetic of independent and identically distributed random variables previously obtained by Hsu Robbins, Erd\H{o}s, Heyde. apply our Gauss map obtain new asymptotics in theorem L\'evy regular continued fraction expansion irrational numbers $(0,1)$.