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

We show that for every ergodic and aperiodic probability preserving system ( X , B , m , T ) (X,B,m,T) , there exists f : X → Z d f:X→ Z^d , whose corresponding cocycle satisfies the d d -dimensional local central limit theorem. We use the 2 2 -dimensional result to resolve a question of Huang, Shao and Ye and Frantzikinakis and Host regarding non-convergence in L 2 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.