On Additive Complements

Two infinite sequnences A and B of non-negative integers are called additive complements,if their sum A+B={a+b:a ∈A,b∈B} contains all sufficiently large integers.Let A(x)and B(x)be the counting functions of A and B.In 2014,Chen and Fang proved that for any additive complements A and B,if then A(x)B(x)-x→+∞,as x→+∞.In 2016,Liu and Fang proved that for any integers a,b with 2≤a≤b,there exists additive complements A and B such that and A(x)B(x)-x=1 for infinitely positive integers x.In this paper,we extend the results of Liu and Fang and prove the following results.(ⅰ)There exist additive complements A and B such that and A(x)B(x)-x=1 for infinitely positive integers x.(ⅱ)There exist additive complements A and B such that and A(x)B(x)-x=1 for infinitely positive integers x.Let(?):A and B are additive complements and A(x)B(x)-x=1 for infinitely-positive integers x},and L’ be the derived set of L,that is,the set of all cluster points of L.(ⅲ)For any integer a,b with b>a≥2 and b≥a+2,we have 2/(1+a/b(a+1))∈L’.