Erdős Problem #763

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 The error term here was improved to \(o(N^{1/4})\) by Jurkat (unpublished) and Montgomery and Vaughan [MoVa90]. See also [764](https://www.erdosproblems.com/764) for a generalisation to more summands.