Random constructions for translates of non-negative functions

Abstract

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $\Lambda$ is type $2$ if the series $s(x)={\sum }_{\lambda \in \Lambda }f(x+\lambda )$ does not satisfy a zero-one law. This means that we can find a non-negative measurable "witness function" $f:\mathbb{R}\to [0,+\infty )$ such that both the convergence set $C(f,\Lambda )=\{x:s(x)<+\infty \}$ and its complement the divergence set $D(f,\Lambda )=\{x:s(x)=+\infty \}$ are of positive Lebesgue measure. If $\Lambda$ is not type $2$ we say that $\Lambda$ is type $1$. 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 $\Lambda$ 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 ${\sum }_{n=1}^{\infty }{c}_{n}f(x+{\lambda }_n)$.