Problem Statement
Let $A\subset \mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B$ and integer $t$ such that\[B+B+t\subseteq A?\]
Categories:
Number Theory Additive Combinatorics
Progress
Erdős [Er75b] posed this as a candidate for a density version of Hindman's theorem (see [172]).This is true, and was proved by Kra, Moreira, Richter, and Robertson [KMRR24].
See also [109].
Source: erdosproblems.com/656 | Last verified: January 16, 2026