On Series of Translates of Positive Functions. III

Abstract

Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is of type 1 if the series $$s(x) = \sum\nolimits_{\lambda \in \wedge } {f(x + \lambda )} $$ satisfies a zero-one law. This means that for any non-negative measurable f: ℝ → [0,+∞) either the convergence set C(f, Λ) = {x: s(x) < +∞} = ℝ modulo sets of Lebesgue zero, or its complement the divergence set D(f, Λ) = {x: s(x) = +∞} = ℝ modulo sets of measure zero. If Λ is not of type 1 we say that Λ is of type 2. We show that there is a universal Λ with gaps monotone decreasingly converging to zero such that for any open subset G ⊂ ℝ one can find a characteristic function f G such that G ⊂ D(f G , Λ) and C(f G , Λ) = ℝ\G modulo sets of measure zero. We also consider the question whether C(f,Λ) can contain non-degenerate intervals for continuous functions when D(f, Λ) is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.