DENSE SETS OF INTEGERS WITH A PRESCRIBED REPRESENTATION FUNCTION
Min Tang · Bulletin of the Australian Mathematical Society · 2011
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.