On additive properties of general sequences

Abstract

LetA= {a1,a2,…} (a1<a2< …) be an infinite sequence of positive integers. LetA(n) be the number of elements ofAnot exceedingn, and denote byR2(n) the number of solutions ofai+aj=n, i≤j. In 1986, Erdős, Sárközy and Sós proved that if (n−A(n))/logn→ ∞(n→ ∞), then. In this paper, we generalise this theorem and give its quantitative form. For example, one of our conclusions implies that if limsup(n−A(n))/logn= ∞, thenfor infinitely many positive integersN.