A strengthening of a theorem of Bourgain-Kontorovich-IV (1503.06132v2)

Abstract: Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator of a finite continued fraction b/d = [d1,d2,...,dk], with all partial quotients d1,d2,...,dk being bounded by an absolute constant A. Several new theorems concerning this conjecture were proved by Bourgain and Kontorovich in 2011. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A = 50 has positive proportion in natural numbers. In 2014 I. D. Kan and D. A. Frolenkov proved this result with A = 5. In this paper the same theorem is proved with A = 4.