Zaremba's conjecture (1971) states that every positive integer number can be represented as a denominator (continuant) of a finite continued fraction with all partial quotients being bounded by an absolute constant A. 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. In 2013 we proved this result with A=7. In this paper the same theorem is proved with A=5.
We prove that the set of positive integers contains a positive proportion of numbers satisfying Zaremba’s conjecture withA= 7. This result strengthens a similar theorem of Bourgain and Kontorovich obtained for A= 50.
We prove that positive integers contains a positive proportion of numbers, satisfying Zaremba's conjecture with A=7. This result is a strengthening of a similar theorem of Bourgain-Kontorovich, obtained for A=50.
Full papers (articles) of 2nd Stochastic Modeling Techniques and Data Analysis (SMTDA-2012) International Conference are represented in the proceedings. This conference took place from 5 June by 8 June 2012 in Chania, Crete, Greece.