Strengthening of the Burgein--Kontorovich theorem on small values of Hausdorff dimension

I. D. Kan

Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 66-80

Résumé

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 $|\mathfrak{D}_{\{1,2 \}}(N)|$ $\gg N^{0.531+0.006}$.

Détail

Keywords: continued fraction, trigonometric sum
Mots-clés : Zaremba's conjecture, Hausdorff dimension.