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)=∑λ∈Λf(x+λ) does not satisfy a zero-one law. This means that we can find a non-negative measurable “witness function” f:R→[0,+∞) such that both the convergence set C(f,Λ)={x:s(x)<+∞} and its complement the divergence set D(f,Λ)={x:s(x)=+∞} 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 ∑cnf(nx) and the Khinchin conjecture, we also discuss some results about weighted sums∑n=1∞cnf(x+λn).