A converse to a theorem of Erdös and Fuchs (Q676231)

In the paper the following is proved: There exists a non-decreasing sequence \(\{a_k, k\geq 0\}\) of nonnegative integers such that the function \(S(n)= \text{card}\{(i,j): a_i+a_j\leq n\}\) satisfies \[ S(n)= cn+O(n^{1/4}\log n)\quad\text{as }n\to\infty. \] The probabilistic proof of this result is based on a Bernstein-type inequality for bounded random variables.