On additive representation functions

Abstract

Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A,n)$ denote the number of solutions of the equation $n=a+b$ with $a,b\in A,a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal to $x$. In this paper, we prove that, if $r(A,n)\ne 1$ for all sufficiently large integers $n$, then $|A(x)|>\frac{1}{2}(\log x/\log \log x{)}^2$ for all sufficiently large $x$.

10 pages