Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

Abstract

We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the $2$-dimensional result to 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. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.