  We investigate the dynamics of continued fractions and explore the ergodic
behaviour of the products of mixed partial quotients in continued fractions of
real numbers. For any function $\Phi:\mathbb N\to [2,+\infty)$ and any integer
$d\geq 1$, we determine the Lebesgue measure and Hausdorff dimension of the set
of real numbers for which the product of partial quotients in arithmetic
progressions satisfy $a_n(x)a_{2n}(x)\cdots a_{dn}(x)\geq \Phi(n)$ for
infinitely many positive integers $n$.
  Our findings shed light on the size of the set of exceptions to Bourgain's
(1988) and Host and Kra's (2005) theorems concerning the convergence of
multiple ergodic averages for Gauss dynamical systems. By exploring the
Hausdorff dimension of these sets, we gain valuable insights into the behaviour
of such exceptions. Overall, our research contributes to a deeper understanding
of the dynamics of continued fractions and their connection to the convergence
properties of ergodic averages in Gauss dynamical systems.
