= Furstenberg multiple recurrence theorem
{c}
{title2=$\mu(\bigcap_{j=0}^{k-1}T^{-jn}B)>0$}
= Multiple recurrence theorem
{synonym}
For a probability <measure-preserving system>, every measurable $B$ of positive measure and every integer $k\geq2$ admit an integer $n\geq1$ 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>.
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