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

TL;DR

This paper investigates the properties and classifications of discrete sets of nonnegative real numbers, called type 1 and type 2, based on a zero-one law for series of translated functions, providing new characterizations and exploring their algebraic and set-theoretic properties.

Contribution

It offers a complete characterization of a subclass of dyadic type 1 sets and explores the structure of type 1 and 2 sets, including their subsets, unions, and sums.

Findings

Characterization of a subclass of dyadic type 1 sets.

Existence of type 1 sets with infinitely many algebraically independent elements.

Analysis of unions and Minkowski sums of type 1 and 2 sets.

Abstract

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is type $1$ if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {{\mathbb R}}\to [0,+ {\infty})$ either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= {{\mathbb R}}$ 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 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.