If $A$ is an additive basis of order $2$, and $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$, then must $A$ contain a minimal additive basis of order $2$? (i.e. such that deleting any element creates infinitely many $n\notin A+A$)What if $1_A\ast 1_A(n)>ϵ\log n$ (for all large $n$, for arbitrary fixed $ϵ>0$)?

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

Write the (unordered) representation function as [ r_A(n):=|\\{(a,a')\in A^2:\ a\le a',\ a+a'=n\\}|. ] Then (1_A*1_A(n)) counts **ordered** pairs, so [ r_A(n)\ \le\ 1_A*1_A(n)\ \le\ 2,r_A(n) ] (up to the diagonal), and in particular “(\to\infty)” and “(\gg \log n)” are unaffected by switching between these two conventi…

llm-hunter · gpt pro 5.2 · unverified

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