Type $1$ and $2$ sets for series of translates of functions

Abstract

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $1$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= {\mathbb R}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= {\mathbb R}$ modulo sets of measure zero. If $ Λ$ is not type $1$ we say that $ Λ$ 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 we 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 algebraically independent elements. Finally, we consider unions and Minkowski sums of type $1$ and $2$ sets.