On the convergence of ∑ n = 1 ∞ f ( n x ) for measurable functions

Questions of Haight and of Weizsäcker are answered in the following result. There exists a measurable function f: (0, + ∞) → {0,1} and two non‐empty intervals I F I ∞ ⊂[½,1) such that∑ n = 1 ∞ f ( n x )= + ∞for every x εI ∞ , and∑ n = 1 ∞ f ( n x )< + ∞ for almost every x εI f . The function f may be taken to be the characteristic function of an open set E .