On the convergence of Σn=1∞f(nx) for measurable functions

Questions of Haight and of Weizsacker are answered in the following result. There exists a measurable function f: (0, +infinity)-->{0, 1} and two non-empty intervals I-F, I-infinity subset of [1/2, 1) such that Sigma (infinity)(n = 1),f (nx) = +infinity for every x is an element ofI(infinity), and Sigma (infinity)(n = 1)f(nx) < +<infinity> for almost every x is an element ofI(F). The function f may be taken to be the characteristic function of an open set E.