On an additive representation function

AbstractLet A be an infinite subset of natural numbers, n∈N and X a positive real number. Let r(n) denotes the number of solution of the equation n=a1+a2 where a1⩽a2 and a1, a2∈A. Also let |A(X)| denotes the number of natural numbers which are less than or equal to X and belong to A. For those A which satisfy the condition that for all sufficiently large natural numbers n we have r(n)≠1, we improve the lower bound of |A(X)| given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible