We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, ErdH
{o}s, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Levy on the regular continued fraction expansion of irrational numbers in $(0,1)$.
We study the run length function for intermittency maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense
that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittency maps. Our results are compensational to the ErdH{o}s-R{e}nyi law obtained by Denker and Nicol in cite{dennic13}.
Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.
