Generalization of a problem of Evelyn-Linfoot and Page in additive number theory

Summary Let k and l be integers such that 2k l. Let S_k and S_llbe two subsets of positive integers with S_kQ_k and S_l Q_l, where Q_k denotes the set of k-free integers. Further suppose that the characteristic functions of S_k and S_ll are multiplicative. Let T(n) denote the number of representations of n in the form n=a+b, where a S_k and b S_ll. In this paper we establish an asymptotic formula for T(n), when n is sufficiently large; and deduce several known asymptotic formulae as particular cases.