Type 1 and 2 sets for series of translates of functions

Abstract

Suppose $${\Lambda}$$ is a discrete infinite set of nonnegative real numbers. We say that $${\Lambda}$$ is type 1 if the series $$s(x)=\sum\nolimits_{\lambda\in\Lambda}f(x+\lambda)$$ satisfies a “zero-one” law. This means that for any non-negative measurable $$f \colon \mathbb{R} \to [0,+ {\infty})$$ either the convergence set $$C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= \mathbb{R}$$ modulo sets of Lebesgue zero, or its complement the divergence set $$D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= \mathbb{R}$$ modulo sets of measure zero. If $${\Lambda}$$ is not type 1 we say that $${\Lambda}$$ is type 2. The exact characterization of type 1 and type 2 sets is not known. In this paper we continue our study of the properties of type 1 and 2 sets. We discuss sub and supersets of type 1 and 2 sets and give a complete and simple characterization of a subclass of dyadic type 1 sets. We discuss the existence of type 1 sets containing infinitely many elements independent over the rationals. Finally, we consider unions and Minkowski sums of type 1 and 2 sets.