Construction of sets of positive measure not containing an affine image of a given infinite structure
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.
infinite version of the 3- dimensional Szemeredi phenomenon
scientific article; zbMATH DE number 4055955
scientific article