On additive properties of general sequences

Let A = {a(1),a(2),...} (a(1) < a(2) < (...)) be an infinite sequence of positive integers. Let A(n) be the number of elements of A not exceeding n, and denote by R-2(n) the number of solutions of a(i) + a(j) = n, i <= j. In 1986, Erdos, Sarkozy and Sos proved that if (n - A(n))/log n -> infinity(n -> infinity), then lim sup (k=1)Sigma(N)(R-2(2k) - R-2(2k + 1)) = +infinity. In this paper, we generalise this theorem and give its quantitative form. For example, one of our conclusions implies that if lim sup (n - A(n)) /log n = infinity, then (n <= N2)max (k=1)Sigma(n)(R-2 (2k) - R-2(2k+1)) >= 0.004 min{A(N), (N-A(N))/log N} for infinitely many positive integers N.