Construction of sets of positive measure not containing an affine image of a given infinite structure
type: Item
instance of: scholarly article
title: Construction of sets of positive measure not containing an affine image of a given infinite structure
zbMATH Open document ID: 0647.28001
DOI: 10.1007/BF02780397
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: A subset S of \({\mathbb{R}}^ d\) has property (E) provided any set in \({\mathbb{R}}^ d\) of positive measure contains an ``affine'' image of S (only translations and homotheties are considered). Erdős asked the question whether there exists an infinite set in \({\mathbb{R}}\) with property (E). In this paper the author shows that if S is an infinite subset of \({\mathbb{R}}\), then the set \(S\times S\times S\) in \({\mathbb{R}}^ 3\) fails to have property (E). By similar arguments, the author says that he can prove that \(S_ 1+S_ 2+S_ 3\) fails (E) for arbitrary infinite subsets \(S_ 1\), \(S_ 2\), \(S_ 3\) of \({\mathbb{R}}\). In the proof, property (E) is reformulated into some integral inequality. Then a rather intricate probabilistic construction gives the result.
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 4055955
zbMATH Keywords: affine images of measurable subsets
zbMATH Keywords: infinite version of the 3- dimensional Szemeredi phenomenon
author: Jean Bourgain
reviewed by: V. Losert
MaRDI profile type: Publication
cites work: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions
cites work: An ergodic Szemerédi theorem for commuting transformations
cites work: On sets of integers containing k elements in arithmetic progression
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description: scientific article; zbMATH DE number 4055955