Geodesic
A~strengthening of the Bourgain-Kontorovich method: three new theorems
Sbornik. Mathematics, Tome 212 (2021) no. 7, pp. 921-964

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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.
Keywords: continued fraction, trigonometric sum
Mots-clés : Zaremba's conjecture, Hausdorff dimension. 
@article{SM_2021_212_7_a1,
     author = {I. D. Kan},
     title = {A~strengthening of the {Bourgain-Kontorovich} method: three new theorems},
     journal = {Sbornik. Mathematics},
     pages = {921--964},
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     number = {7},
     year = {2021},
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     url = {http://geodesic.mathdoc.fr/item/SM_2021_212_7_a1/}
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I. D. Kan. A~strengthening of the Bourgain-Kontorovich method: three new theorems. Sbornik. Mathematics, Tome 212 (2021) no. 7, pp. 921-964. http://geodesic.mathdoc.fr/item/SM_2021_212_7_a1/