A reinforcement of the Bourgain-Kontorovich's theorem by elementary methods

Abstract: 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 .