A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,μ a Borel probability measure on X , T : X → X a measure preserving continuous transformation and g : X → R a continuous function. Then for some y ∈ X , ∀ N ⩾ 1 1 N ∑ n = 1 N g ( T n ( y ) ) ⩾ ∫ g   d μ.This Lemma is used to give an alternative proof of a result by Ruzsa [6], which implies the following extension of a result of Bergelson [1]. If E ⊂ N satisfiesB D ¯ ( E ) = inf n > 0sup k ⩾ 01 n | E ∩ [ k + 1 ,   k + n ]| ⩾ 0 , then there exists a set Λ ⊂ N such that n −1|Λ∩[1,n]| ⩾B D ¯ (E) for all, n ⩾ 1, and any finite subset {λ 1 , … λ k } ⊂ Λ satisfies∩ i = 1 k( E + λ i )≠ Ø.