Random constructions for translates of non-negative functions

Suppose $\mathrm{\Lambda }$ is a discrete infinite set of nonnegative real numbers. We say that $\mathrm{\Lambda }$ is type $2$ if the series $s(x)={\sum }_{\lambda \in \mathrm{\Lambda }}f(x+\lambda )$ does not satisfy a zero-one law. This means that we can find a non-negative measurable “witness function” $f:\mathbb{R}\to [0,+\mathrm{\infty })$ such that both the convergence set $C(f, {\Lambda})={x: s(x)<+ {\infty} }$anditscomplementthedivergenceset$D(f, {\Lambda})={x: s(x)=+ {\infty} }$areofpositiveLebesguemeasure\mathrm{.}If$ {\Lambda}$isnottype$2$wesaythat$ {\Lambda}$istype$1$. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set $\mathrm{\Lambda }$ if we randomly delete its elements. Motivated by results concerning weighted sums $\sum c_{n}f(nx)$ and the Khinchin conjecture, we also discuss some results about weighted sums ${\sum }_{n=1}^{\mathrm{\infty }}{c}_{n}f(x+{\lambda }_n)$.