METRICAL PROPERTIES FOR THE WEIGHTED PRODUCTS OF MULTIPLE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

The classical Khintchine and Jarnik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grow at a certain rate. Recently it was observed that the growth of the product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set: epsilon(t)(psi) := {x is an element of[0, 1) : Pi(m-1)(i=0) a(n+i)(ti) (x) >= Psi(n) for infinitely many n is an element of N}, where t(i) is an element of R+ for all 0 <= i <= m - 1, and Psi: N -> R->= 1 is a positive function.