Metrical properties for functions of consecutive multiple partial quotients in continued fractions.

Recently, the growth of the products of consecutive partial quotients a i (x) in the continued fraction expansion of a real number x was studied in connections with improvements to Dirichlet's theorem. In this paper, for a non-decreasing positive measurable function F (x 1 , ... , x m) and a function ϕ : ℕ → ℝ > 0 , we consider the set ℰ F (ϕ) = { x ∈ [ 0 , 1 ] : F (a n (x) , ... , a n + m − 1 (x)) ≥ ϕ (n)  for infinitely many  n ∈ ℕ } , and obtain its Lebesgue measure ℒ (ℰ F (ϕ)). As an application of our result, we reprove a theorem of Bakhtawar–Hussain–Kleinbock–Wang. We also consider the case when F (x 1 , ... , x m) = x 1 + ⋯ + x m . [ABSTRACT FROM AUTHOR]