Furstenberg multiple recurrence theorem

Statement

For a probability-preserving transformation $T$ , a measurable set $A$ with $\mu(A)>0$ , and any integer $k\geq1$ , there exists $n\geq1$ such that

\mu(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-kn}A)>0.

This is Furstenberg's multiple recurrence theorem. Ergodicity is not required, and the statement also holds for noninvertible transformations.

Koopman formulation

For forward composition $Uf=f\circ T$ , the measure above equals

\int_X\prod_{j=0}^{k}U^{jn}1_A\,d\mu.

This is a multiple correlation, involving products of evolved observables. Ordinary two-function correlations alone do not capture this statement.

Arithmetic application

The correspondence principle turns positive density in the integers into positive probability in a shift system. Multiple recurrence then yields Szemerédi's theorem on arithmetic progressions.

References

Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. Theorem 1.1.7 and Lecture 10.