For any T measure-preserving, K ≥ 1, and A with μ(A) > 0, there is n with μ(A ∩ T^{-n}A ∩ ⋯ ∩ T^{-(K-1)n}A) > 0. Furstenberg's correspondence principle makes this equivalent to Szemerédi's theorem on arithmetic progressions.
For any T measure-preserving, K ≥ 1, and A with μ(A) > 0, there is n with μ(A ∩ T^{-n}A ∩ ⋯ ∩ T^{-(K-1)n}A) > 0. Furstenberg's correspondence principle makes this equivalent to Szemerédi's theorem on arithmetic progressions.