On series of translates of positive functions III

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.