Furstenberg multiple recurrence theorem ($\mu ({\bigcap }_{j=0}^{k-1}{T}^{-jn}B)>0$)

For a probability measure-preserving system, every measurable $B$ of positive measure and every integer $k\ge 2$ admit an integer $n\ge 1$ with $\mu ({\bigcap }_{j=0}^{k-1}{T}^{-jn}B)>0$. The sets are preimages; neither invertibility nor an ergodic transformation is required. The Furstenberg correspondence principle converts this simultaneous return into arithmetic progressions and yields the Szemerédi theorem. The case $k=2$ follows from the Poincaré recurrence theorem.

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.

Every circle rotation has the Furstenberg multiple recurrence theorem property for normalized Lebesgue measure. For a rational angle, use a period. For an irrational angle, choose a positive return time with angle close to zero; translation continuity in L1 on the circle then makes finitely many translates of a given positive-measure set simultaneously close to that set, and the union bound leaves a positive-measure intersection.