Before revision:  We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, regardless of the properties of the remaining (absolutely continuous, or singular) distribution on the positive real line.
After revision:  We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, under mild assumptions on the remaining distribution on the positive real line.