SZ property (${\mathrm{liminf}}_N{N}^{-1}{\sum }_{n=1}^N\mu ({\bigcap }_{j=0}^k{T}^{-jn}A)>0$)

A probability measure-preserving system has the SZ property at level $k$ if the displayed lower limit is positive for every measurable set $A$ of positive measure. This convention counts $k$ nonzero shifts, hence progressions of length $k+1$. The Furstenberg correspondence principle turns this property at all levels into the Szemerédi theorem. A compact measure-preserving system has the property because syndetic near returns of an almost periodic observable applied to an indicator give uniformly positive multiple intersections at bounded-gap times.