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The theory of uniform Diophantine approximation concerns the study ofDirichlet improvable numbers and the metrical aspect of this theory leads tothe study of the product of consecutive partial quotients in continuedfractions. It is known that the dimension of the set of Dirichletnon-improvable numbers depends upon the number of partial quotients in theproduct string. However, one can see that the Hausdorff dimension is the samefor any number of consecutive partial quotients with a constant gap. This paperis aimed at a detailed analysis on how the Hausdorff dimension changes whenthere is a linear gap in indices and the number of partial quotients in theproduct 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 theset\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 thedimension is larger if $d$ is larger and $t$ has no contribution to thedimension.

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