A strengthening of a theorem of Bourgain and Kontorovich. V

I. D. Kan

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and combinatorial number theory, Tome 296 (2017), pp. 133-139

Résumé

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