A strengthening of the Bourgain-Kontorovich method: three new theorems

Abstract

Consider the set of irreducible denominators of the rational numbers representable by finite continued fractions all of whose partial quotients belong to some finite alphabet . Let the set of infinite continued fractions with partial quotients in this alphabet have Hausdorff dimension satisfying . Then contains a positive share of positive integers. A previous similar result of the author of 2017 was related to the inequality 0.7807\\dots$?> ; in the original 2011 Bourgain-Kontorovich paper, 0.9839\\dots$?> . Bibliography: 28 titles.