On the additive completion of squares (Q1103001)

Let \(B\) be a subset of integers. Assume that every positive integer \(n<N\), where \(N\) is a sufficiently large integer, can be written as \(n=b+\lambda^2\) for some \(b\in B\) and some integer \(\lambda\). It is shown that \(\operatorname{card} B>(1.15)\cdot \sqrt{N}\), thus improving a known result. The generalization where \(\lambda^2\) is replaced by \(f(\lambda)\) is pointed out and the case \(f(\lambda)=\lambda^3\) considered.