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In this paper, we investigate the dynamics of continued fractions and explorethe ergodic behaviour of the products of mixed partial quotients. For any$d\geq 1$ and $\Phi:\N\to \R_+$, our focus lies on analysing the Hausdorffdimension of the set of real numbers for which the product of mixed partialquotients $a_n(x)a_{2n}(x)\cdots a_{dn}(x)\geq \Phi(n)$ for infinitely manypositive integers $n$.  Our findings shed light on the size of the set of exceptions to Bourgain(1988) and Host-Kra (2005) theorems concerning the convergence of multipleergodic averages for Gauss dynamical systems. By exploring the Hausdorffdimension of these sets, we gain valuable insights into the behaviour of suchexceptions. Overall, our research contributes to a deeper understanding of thedynamics of continued fractions and their connection to the convergenceproperties of ergodic averages in Gauss dynamical systems.

Metrical properties of finite product of partial quotients in arithmetic progressions