Abstract: In this talk, for an ergodic probability preserving system $(X,\mathcal{B},m,T)$, we will discuss the existence of a function $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. 
As an application, we resolve a question of Huang, Shao and Ye, and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. We also also provide first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff.