 Furstenberg multiple recurrence theorem

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 Poincare recurrence theorem.