On series of translates of positive functions
Let us consider the collection \({\mathcal D}\) of all discrete infinite sets of nonnegative real numbers. For \( \Lambda \in {\mathcal D} \) and a nonnegative Lebesgue measurable function \( f : {\mathbb{R}} \to {\mathbb R}^{+} \) let \( C(f,\Lambda) = \{ x \in {\mathbb R} : \sum_{\lambda \in \Lambda} f(x+\lambda) < +\infty\} \). The collection \({\mathcal D}\) is divided into two types. Type 1 consists of \( \Lambda \in {\mathcal D} \) such that, for every admissible \(f\), either \( C(f,\Lambda) \) is a null set or \( {\mathbb R} \setminus C(f,\Lambda) \) is a null set (in the sense of Lebesgue measure). Type 2 contains the rest of \({\mathcal D}\). Classes of examples for sets of both types are enumerated. For instance, it is demonstrated that \(\Lambda\) is of type 2 if it is asymptotically lacunary or it is asymptotically dense and exhibits asymptotically large \({\mathbb Q}\)-independent sets. It is also established that sets of type 2 form a dense open subset of \({\mathcal D}\), while sets of type 1 form a closed nowhere dense subset of \({\mathcal D}\) in the box topology.
scientific article; zbMATH DE number 1870466