A reinforcement of the Bourgain-Kontorovich's theorem by elementary   methods
  Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 has positive proportion in $\N.$ The proof of this theorem is rather complicated and refers to the spectral theory. In this paper,using only elementary methods, the same theorem is proved with A=13 . 
Dmitriy Frolenkov, Igor D. Kan
math.NT