Random constructions for translates of non-negative functions

Abstract

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $2$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ 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 $C(f, Λ)=\{x: s(x)<+ {\infty} \}$ and its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}$ are of positive Lebesgue measure. If $ Λ$ is not type $2$ we say that $ Λ$ 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 $ Λ$ 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 $\sum_{n=1}^{\infty}c_n f(x+λ_n)$.