On the density of sequences of integers the sum of no two of which is a square II. General sequences

The aim of this paper is to study the maximal density attainable by a sequence S of positive integers having the property that the sum of any two distinct elements of S is never a square. It is shown that there is a constant N₀ such that for all N ? N₀ any set S ? [1, N] having this property must have |S| < 0.475N. The proof uses the Hardy-Littlewood circle method.