A strengthening of a theorem of Bourgain and Kontorovich. V

Abstract

It is proved that the denominators of finite continued fractions all of whose partial quotients belong to an arbitrary finite alphabet A with parameter δ > 0.7807... (i.e., such that the set of infinite continued fractions with partial quotients from this alphabet is of Hausdorff dimension δ with δ > 0.7807... ) contain a positive proportion of positive integers. Earlier, a similar theorem has been obtained only for alphabets with somewhat greater values of δ. Namely, the first result of this kind for an arbitrary finite alphabet with δ > 0.9839... is due to Bourgain and Kontorovich (2011). Then, in 2013, D.A. Frolenkov and the present author proved such a theorem for an arbitrary finite alphabet with δ > 0.8333.... The preceding result of 2015 of the present author concerned an arbitrary finite alphabet with δ > 0.7862....