Strengthening of the Bourgain–Kontorovich Theorem on Small Values of Hausdorff Dimension

Abstract

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