Density theorems revisited
Let \(\varPhi \) be a family of real intervals. A sequence of pairwise disjoint intervals \(\{I_n\}\) is a basis according to \(\varPhi \) if \(I_n\to 0\) and if for a.e.\ \(x\in \mathbb R\setminus \bigcup \varPhi \) there are only finitely many \(n\) such that \(x+I_n\in \varPhi \). Equivalent conditions are given in order that a sequence \(\{I_n\}\) of pairwise disjoint intervals, \(I_n\to 0\), is a universal basis, that is, \(\{I_n\}\) is a basis for every family of real intervals. An application to density theorems for measurable sets in \(\mathbb R\) is given.
scientific article; zbMATH DE number 1449390