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.
Categories: math.CA