Problem Statement
Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density?
Is there a basis $A$ of order $2$ such that if $A=A_1\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?
Is there a basis $A$ of order $2$ such that if $A=A_1\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?
Categories:
Additive Combinatorics
Progress
A problem of Burr and Erdős. Erdős [Er94b] thought he could construct a basis as in the second question, but 'could never quite finish the proof'.Source: erdosproblems.com/741 | Last verified: January 16, 2026