Problem Statement
Is there an explicit construction of a set $A\subseteq \mathbb{N}$ such that $A+A=\mathbb{N}$ but $1_A\ast 1_A(n)=o(n^\epsilon)$ for every $\epsilon>0$?
Categories:
Number Theory Additive Basis
Progress
The existence of such a set was asked by Sidon to Erdős in 1932. Erdős (eventually) proved the existence of such a set using probabilistic methods. This problem asks for a constructive solution.An explicit construction was given by Jain, Pham, Sawhney, and Zakharov [JPSZ24].
Source: erdosproblems.com/29 | Last verified: January 13, 2026