On sets with sum and difference structure

For nonempty sets [Formula: see text] of nonnegative integers and an integer n, let [Formula: see text] be the number of representations of n as [Formula: see text] and [Formula: see text] be the number of representations of n as [Formula: see text], where [Formula: see text]. For a nonnegative real number x, let [Formula: see text] (respectively [Formula: see text]) be the number of elements in A (respectively B) not exceeding x. In this paper, we determine the sets [Formula: see text] such that [Formula: see text] for every nonnegative integer n. We also consider the difference structure and prove that: there exist sets A and B of nonnegative integers such that [Formula: see text] for every nonnegative integer n, [Formula: see text] and [Formula: see text] for every nonnegative integer k. Other related results are also established.