Geodesic
Modular Generalization of the Bourgain--Kontorovich Theorem
Matematičeskie zametki, Tome 114 (2023) no. 5, pp. 739-752

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The set $\mathfrak{D}^N_\mathbf{A}$ of all irreducible denominators $\le N$ of positive rationals $1$ whose continued fraction expansions consist of elements of the set $\mathbf{A}=\{1,2,4\}$ is considered. We prove that, for any prime $Q\le N^{2/3}$, the set $\mathfrak{D}^N_{\mathbf{A}}$ contains almost all possible remainders on division by $Q$ and the remainder term in the corresponding asymptotic formula decays according to a power law.
Keywords: continued fraction, trigonometric sum
Mots-clés : Zaremba's conjecture, Hausdorff dimension. 
@article{MZM_2023_114_5_a6,
     author = {I. D. Kan},
     title = {Modular {Generalization} of the {Bourgain--Kontorovich} {Theorem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {739--752},
     publisher = {mathdoc},
     volume = {114},
     number = {5},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_114_5_a6/}
}
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I. D. Kan. Modular Generalization of the Bourgain--Kontorovich Theorem. Matematičeskie zametki, Tome 114 (2023) no. 5, pp. 739-752. http://geodesic.mathdoc.fr/item/MZM_2023_114_5_a6/