On additive representation function of general sequences (Q2460647)

Let \(0\leq a_1<a_2<\cdots\) be an infinite sequence \(\mathcal A\) of integers and let \(r_1(\mathcal{A},n)=| (i,j):a_i+a_j=n,\;i\leq j| \). It is known that \(r_1(\mathcal{A},n)\) cannot be a constant for sufficient large \(n\). In the note under review the author improves this result as follows: If \(d>0\) is an integer, then there does not exist \(n_0\) such that \[ d\leq r_1(\mathcal{A},n)\leq d+\left[\sqrt{2d}+\frac12\right] \] for \(n>n_0\).