Suppose A is a discrete infinite set of nonnegative real numbers. We say that A is
type 2 if the series s(x) = Sigma lambda Lambda f (x + lambda) does not satisfy a
zero-one law. This means that we can find a non-negative measurable "witness function"
f : R -> [0,+ infinity) such that both the convergence set C(f, Lambda) ={x : s(x)
< + infinity} and its complement the divergence set D (f, Lambda) = {x : s(x) = +infinity}
are of positive Lebesgue measure. If Lambda is not type 2 we say that A 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 A if we randomly delete its elements. Motivated by
results concerning weighted sums Sigma c(n)f(nx)and the Khinchin conjecture, we also
discuss some results about weighted sums
