In this talk, I will talk about the Lebesgue measure and Haus- dorff dimension of the set of real numbers for which the product of partial quotients in arithmetic progressions satisfy an(x)a2n(x) · · · adn(x) ≥ Φ(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.