Geodesic
Strengthening of the Burgein--Kontorovich theorem on small values of Hausdorff dimension
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 66-80.

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Let $\mathfrak{D}_\mathbf{A}(N)$ be the set of all integers not exceeding $N$ and equal to irreducible denominators of positive rational numbers with finite continued fraction expansions in which all partial quotients belong to a finite number alphabet $\mathbf{A}$. A new lower bound for the cardinality $|\mathfrak{D}_\mathbf{A}(N)|$ is obtained, whose nontrivial part improves that known previously by up to $28\%$. Thus, for $\mathbf{A}=\{1,2\}$, a formula derived in the paper implies the inequality $|\mathfrak{D}_{\{1,2 \}}(N)|\gg N^{0.531+0.024}$ with nontrivial part $0.024$. The preceding result of the author was $|\mathfrak{D}_{\{1,2 \}} (N)|\gg N^{0.531+0.019}$, and a calculation by the original 2011 theorem of Bourgain and Kontorovich gave $|\mathfrak{D}_{\{1,2 \}}(N)|$ $\gg N^{0.531+0.006}$.
Keywords: continued fraction, trigonometric sum
Mots-clés : Zaremba's conjecture, Hausdorff dimension. 
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I. D. Kan. Strengthening of the Burgein--Kontorovich  theorem on small values of Hausdorff dimension. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 66-80. http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a4/

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