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

TL;DR

This paper investigates the measure and dimension of sets of real numbers whose continued fraction partial quotients' products grow at a specified rate, extending classical results to multiple partial quotients with different powers.

Contribution

It generalizes the analysis of partial quotient products in continued fractions to multiple consecutive terms with varying powers, providing new measure and dimension results.

Findings

Determines Lebesgue measure of the specified set.

Calculates Hausdorff dimension of the set.

Extends classical Diophantine approximation results.

Abstract

The classical Khintchine and Jarn\'ik 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 grows with a certain rate. Recently it was observed that the growth of 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: $$ {\D_{\mathbf t}}(\psi):=\left\{x\in[0, 1): \prod\limits_{i=0}^{m-1}{a^{t_i}_{n+i}(x)} \ge \Psi(n)\ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $t_i\in\mathbb R_+$ for all ${0\leq i\leq m-1}$, and $\Psi:\N\to\R_{\ge 1}$ is a positive function.