Let $r\ge 2$ and let $A\subseteq \{1,\dots ,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\le b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.)Similarly, let $B\subseteq \{1,\dots ,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n\ge 1$. If $|A|\sim c_r{N}^{1/2}$ as $N\to \infty$ and $|B|\sim c_r^{\prime }{N}^{1/2}$ as $N\to \infty$ then is it true that $c_r\ne c_r^{\prime }$ for $r\ge 2$? Is it true that $c_r^{\prime }<c_r$?

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

A small but important normalization first: in the “difference” problem you must **exclude** $n=0$ [[nomath]](or else require $a\ne b$, equivalently $a>b$ if you only look at positive $n$)[[/nomath]], since (a-b=0) has (|B|) solutions and would force (|B|\le r), contradicting the asserted (\asymp N^{1/2}) growth.

llm-hunter · gpt pro 5.2 · unverified

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