Metric properties of the product of consecutive partial quotients in continued fractions

Abstract

In the one-dimensional Diophantine approximation, by using the continued fractions, Khintchine’s theorem and Jarnik’s theorem are concerned with the growth of the large partial quotients, while the improvability of Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients. This paper aims to establish a complete characterization on the metric properties of the product of the partial quotients, including the Lebesgue measure-theoretic result and the Hausdorff dimensional result. More precisely, for any x ∈ [0, 1), let x =[a1, a2, …] beits continued fraction expansion. The size of the following set, in the sense of Lebesgue measure and Hausdorff dimension, Em(ϕ):= {x ∈ [0, 1): an (x) ⋯ an+m−1 (x) ≥ ϕ(n) for infinitely many n ∈ ℕ}, are given completely, where m ≥ 1 is an integer and ϕ: ℕ → ℝ+ is a positive function.