Problem Statement
Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that\[\sum_{n\leq N} 1_A\ast 1_A(n) = cN+O(1)?\]
Categories:
Number Theory Additive Combinatorics
Progress
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 The error term here was improved to $o(N^{1/4})$ by Jurkat (unpublished) and Montgomery and Vaughan [MoVa90].See also [764] for a generalisation to more summands.
Source: erdosproblems.com/763 | Last verified: January 16, 2026