Metrical properties of the product of partial quotients with geometric mean in continued fractions

TL;DR

This paper investigates how the Hausdorff dimension of certain sets related to continued fractions changes with linear gaps in indices and growth of partial quotient products, revealing that larger gaps increase the dimension.

Contribution

It provides a detailed Hausdorff dimension analysis for sets defined by products of partial quotients with linear index gaps, highlighting the effect of the gap size.

Findings

Dimension increases with larger gap parameter d

Gap parameter t does not affect the Hausdorff dimension

The analysis extends understanding of Diophantine approximation sets

Abstract

The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable numbers and the metrical aspect of this theory leads to the study of the product of consecutive partial quotients in continued fractions. It is known that the dimension of the set of Dirichlet non-improvable numbers depends upon the number of partial quotients in the product string. However, one can see that the Hausdorff dimension is the same for any number of consecutive partial quotients with a constant gap. This paper is aimed at a detailed analysis on how the Hausdorff dimension changes when there is a linear gap in indices and the number of partial quotients in the product grows. More precisely, let $d\in \N_{\ge 1}, t\in\Z_{\geq 0}$ and $f(n)=dn+t$, we present the detailed Hausdorff dimension analysis of the set\begin{equation*} E_{f}(\psi):=\left\{x\in [0, 1): \sqrt[n]{a_{f(n)}(x)a_{2f(n)}(x)\cdots a_{nf(n)}(x)}\geq \psi(n) \ {\rm for \ infinitely \ many} \ n\in \N\right\}. \end{equation*} It is seen that the dimension is larger if $d$ is larger and $t$ has no contribution to the dimension.