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A combinatorial application of the maximal ergodic theorem. (English) Zbl 0642.10051

Let X be a compact space, \(\mu\) a Borel probability measure on X, \(T: X\to X\) a measure preserving transformation and \(g: X\to {\mathbb{R}}^ a \)continuous function. Then for some \(y\in X,\forall N\geq 1\) \[ (1/N)\sum^{n}_{n=1}g[T^ n(y)]\geq \int g d\mu. \] This Lemma is used to give an alternative proof of a result of I. Z. Ruzsa [Stud. Sci. Math. Hung. 13, 319-326 (1978; Zbl 0423.10027)], which implies the following extension of a result of V. Bergelson [J. Lond. Math. Soc., II. Ser. 31, 295-304 (1985; Zbl 0579.10029)]: If \(E\subset {\mathbb{N}}\) satisfies \(\overline{BD}(E)=\inf_{n>0}\sup_{k\geq 0}(1/n)| E\cap [k+1,k+n]| \geq 0,\) (upper Banach density of E) then there exists a set \(\Lambda \subset {\mathbb{N}}\) such that \((1/n)| \Lambda \cap [1,n]| \geq \overline{BD}(E)\) for all \(n\geq 1,\) and any finite subset \(\{\lambda_ 1,\dots,\lambda_ k\}\subset \Lambda\) satisfies \(\cap^{k}_{i=1}(E+\lambda_ i)\neq \emptyset.\)
Reviewer: Y.Peres

MSC:

11B83 Special sequences and polynomials
28D05 Measure-preserving transformations
11B75 Other combinatorial number theory
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