On function $SX$ of additive complements

Two sets A, B of nonnegative integers are called additive complements, if all sufficiently large integers can be expressed as the sum of two elements from A and B. We call A, B perfect additive complements if every nonnegative integer can be uniquely expressed as the sum of two elements from A and B. Let $A(x)$ be the counting function of A. In this paper, we focus on the function $SX$, where SX=x→∞max(x),B(x)\√x was introduced by Erd os and Freud in 1984. As a main result, we determine the value of $SX$ for perfect additive complements, and further fix the infimum. We also give the absolute lower bound of $SX$ for additive complements.