SOLVED

Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that
\[\sum_{n\leq N} 1_A\ast 1_A(n) = cN+O(1)?\]

A conjecture of Erdős and Turán. Erdős and Fuchs [ErFu56] proved that the answer is no in a strong form: in fact even
\[\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)\]
is impossible. The error term here was improved to $N^{1/4}$ by Jurkat (unpublished) and Montgomery and Vaughan [MoVa90].