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

Recently, the growth of the products of consecutive partial quotients ai(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 (x1,... , xm) and a function phi : N -> Ri0, we consider the set EF (phi) = {x is an element of [0,1] : F (a(n)(x), ... , an+m-1(x)) > phi(n) for infinitely many n is an element of N}, and obtain its Lebesgue measure .C(EF(phi)). As an application of our result, we reprove a theorem of Bakhtawar-Hussain-Kleinbock-Wang. We also consider the case when F (x1, ... , xm) = x1 + <middle dot><middle dot><middle dot> + x(m).