If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\mathrm{limsup}{1}_A\ast 1_A(n)=\infty$.

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

Let [ r_A(n):=(1_A*1_A)(n)=\sum_{k\ge 0}1_A(k)1_A(n-k) ] [[nomath]](the number of **ordered** pairs $(a,b)\in A^2$ with $a+b=n$)[[/nomath]]. Your hypothesis “$A+A$ contains all but finitely many integers” is exactly [ r_A(n)>0\qquad\text{for all }n\ge n_0 . ]

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

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