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

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%.