Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that$$d(A+B)=d(A)+d(B).$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Assume throughout that **natural density** exists: [ d(A)=\lim_{N\to\infty}\frac{|A\cap[1,N]|}{N}. ] Write (A\sim A') if (d(A\triangle A')=0) [[nomath]](they differ on a density–$0$ set)[[/nomath]]. Note that (d(A+t)=d(A)) for each fixed $t$, and finite changes don’t affect density.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.