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

Abstract

Recently, the growth of the products of consecutive partial quotients [Formula: see text] in the continued fraction expansion of a real number [Formula: see text] was studied in connections with improvements to Dirichlet’s theorem. In this paper, for a non-decreasing positive measurable function [Formula: see text] and a function [Formula: see text], we consider the set [Formula: see text], and obtain its Lebesgue measure [Formula: see text]). As an application of our result, we reprove a theorem of Bakhtawar–Hussain–Kleinbock–Wang. We also consider the case when [Formula: see text].