On series of translates of positive functions III

TL;DR

This paper investigates the zero-one law for series of translates of positive functions, establishing the existence of a universal set with specific divergence properties and exploring the structure of convergence sets for continuous functions.

Contribution

It introduces a universal set with decreasing gaps where divergence sets can be prescribed and addresses the structure of convergence sets for continuous functions.

Findings

Existence of a universal set with decreasing gaps where divergence sets can be arbitrarily prescribed.

Demonstration that convergence sets can be complements of arbitrary open sets modulo measure zero.

Insights into whether convergence sets can contain non-degenerate intervals for continuous functions.

Abstract

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of 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 of type 1 we say that $ {\Lambda}$ is of type 2. In this paper we show that there is a universal $ {\Lambda}$ with gaps monotone decreasingly converging to zero such that for any open subset $G \subset { {\mathbb R}}$ one can find a characteristic function $f_{G}$ such that $G \subset D(f_G, {\Lambda})$ and $C(f_G, {\Lambda})= { {\mathbb R}} {\setminus} G$ modulo sets of measure zero. We also consider the question whether $C(f, {\Lambda})$ can contain non-degenerate intervals for continuous functions when $D(f, {\Lambda})$ is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.