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

Consider the set $\mathfrak{D}_{\mathbf{A}}$ of irreducible denominators of the rational numbers representable by finite continued fractions all of whose partial quotients belong to some finite alphabet $\mathbf{A}$. Let the set of infinite continued fractions with partial quotients in this alphabet have Hausdorff dimension $\Delta_{\mathbf{A}}$ satisfying $\Delta_{\mathbf{A}} \geqslant0.7748\dots$ . Then $\mathfrak{D}_{\mathbf{A}}$ contains a positive share of positive integers. A previous similar result of the author of 2017 was related to the inequality $\Delta_{\mathbf{A}} >0.7807\dots$; in the original 2011 Bourgain-Kontorovich paper, $\Delta_{\mathbf{A}} >0.9839\dots$ .
Bibliography: 28 titles.