Strengthening of the Bourgain–Kontorovich Theorem on Small Values of Hausdorff Dimension
Игорь Давидович Кан
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 $$|\mathfrak{D}_{\{1,2 \}}(N)|$$ $$\gg N^{0.531+0.006}$$ .