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The classical Khintchine and Jarn\'ik theorems, generalizations of aconsequence of Dirichlet's theorem, are fundamental results in the theory ofDiophantine approximation. These theorems are concerned with the size of theset of real numbers for which the partial quotients in their continued fractionexpansions grows with a certain rate. Recently it was observed that the growthof product of pairs of consecutive partial quotients in the continued fractionexpansion of a real number is associated with improvements to Dirichlet'stheorem. In this paper we consider the products of several consecutive partialquotients raised to different powers. Namely, we find the Lebesgue measure andthe Hausdorff dimension of the following set: $$ {\D_{\mathbft}}(\psi):=\left\{x\in[0, 1): \prod\limits_{i=0}^{m-1}{a^{t_i}_{n+i}(x)} \ge\Psi(n)\ {\text{for infinitely many}} \ n\in \N \right\}, $$ where$t_i\in\mathbb R_+$ for all ${0\leq i\leq m-1}$, and $\Psi:\N\to\R_{\ge 1}$ isa positive function.

Metrical properties for the weighted products of multiple partial quotients in continued fractions