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A Combinatorial Application of the Maximal Ergodic Theorem
Author(s) -
Peres Yuval
Publication year - 1988
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/20.3.248
Subject(s) - mathematics , ergodic theory , probability measure , combinatorics , measure (data warehouse) , lemma (botany) , compact space , space (punctuation) , discrete mathematics , borel measure , extension (predicate logic) , function (biology) , measurable function , pure mathematics , mathematical analysis , bounded function , ecology , linguistics , philosophy , poaceae , database , computer science , biology , evolutionary biology , programming language
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.