A question of Ben Green asks whether every finite set $A$ of integers with doubling constant $K$ must contain a subset $A'$ of comparable size whose doubling is at most $K + o(1)$ due to some explicit algebraic structure on $A'$. This was previously understood in the regime $K < 4 - o(1)$ by work of Eberhard, Green, and Manners, who showed that one can find such a subset $A'$ with density at least $1/2 + o(1)$ inside a long arithmetic progression. In this talk, I will provide a brief survey of this question as well as mention some new progress towards this. This is joint work with Yifan Jing.