Strongly mixing systems are almost strongly mixing of all orders

Vitaly Bergelson, Rigoberto Zelada

Published 2020-10-13Version 1

We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $\mathcal R$-limits, a notion of convergence which is based on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of IP$^*$ and uniform density one. We also demonstrate the versatility of this new approach by obtaining applications to higher order mixing properties of weakly and mildly mixing systems.