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 Zᵈ, 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 L² 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.