Almost Everywhere Convergence Questions of Series of Translates of Non-Negative Functions

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis Paris. line of research was initiated by question Haight and Weizsäker concerning almost everywhere convergence properties series form $\sum_{n=1}^{{\infty}}f(nx)$. A more general, additive version this problem following: Suppose $\Lambda$ discrete infinite set nonnegative real numbers. We say that $ {\Lambda}$ type 1 if $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies zero-one law. means for any non-negative measurable $f: \mathbb R\to [0,+ {\infty})$ either $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= R$ modulo sets Lebesgue zero, or its complement divergence $D(f, s(x)=+ measure zero. If not we 2. The exact characterization $1$ $2$ still known. part discussing results several joint papers written beginning with J-P. Kahane D. Mauldin, later B. Hanson, Maga G. Vértesy. Apart from above project also cover historic background, other related open questions.