= SZ property
{c}
{title2=$\liminf_NN^{-1}\sum_{n=1}^N\mu(\bigcap_{j=0}^kT^{-jn}A)>0$}
= Szemerédi multiple recurrence property
{c}
{synonym}
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.
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