On series of translates of positive functions III

Suppose $\mathrm{\Lambda }$ is a discrete infinite set of nonnegative real numbers. We say that $\mathrm{\Lambda }$ is of type 1 if the series $s(x)={\sum }_{\lambda \in \mathrm{\Lambda }}f(x+\lambda )$ satisfies a zero-one law. This means that for any non-negative measurable $f: { {\mathbb R}}\to [0,+ {\infty})$eithertheconvergenceset$C(f, {\Lambda})={x: s(x)<+ {\infty} }= { {\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f,\mathrm{\Lambda })=\{x:s(x)=+\mathrm{\infty }\}=\mathbb{R}$ modulo sets of measure zero. If $\mathrm{\Lambda }$ is not of type 1 we say that $\mathrm{\Lambda }$ is of type 2. In this paper we show that there is a universal $\mathrm{\Lambda }$ with gaps monotone decreasingly converging to zero such that for any open subset $G \subset { {\mathbb R}}$onecanfindacharacteristicfunction$f_{G}$ such that $G\subset D(f_G,\mathrm{\Lambda })$ and $C(f_G, {\Lambda})= { {\mathbb R}} {\setminus} G$ modulo sets of measure zero. We also consider the question whether $C(f,\mathrm{\Lambda })$ can contain non-degenerate intervals for continuous functions when $D(f,\mathrm{\Lambda })$ is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.