Abstract A set A ⊆ℤ is called an asymptotic basis of ℤ if all but finitely many integers can be represented as a sum of two elements of A . Let A be an asymptotic basis of integers with prescribed representation function, then how dense A can be? In this paper, we prove that there exist a real number c >0 and an asymptotic basis A with prescribed representation function such that $A(-x,x)\geq c\sqrt {x}$ for infinitely many positive integers x .