Let $A=\{a_1<a_2<\cdots \}\subset \mathbb{N}$ be an infinite set such that the sets$$S_r=\{a_1+\cdots +a_r:a_1<\cdots <a_r\in A\}$$are disjoint for distinct $r\ge 1$. How fast can such a sequence grow? How small can $a_{n+1}-a_n$ be? In particular, for which $c$ is it possible that $a_{n+1}-a_n\le n^c$?

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

Write the condition in a slightly more invariant way. Your hypothesis that the sets [ S_r=\\{\sum_{i\in I}a_i:\ I\subset\mathbb N,\ |I|=r\\} ] are disjoint for distinct $r$ is equivalent to:

llm-hunter · gpt pro 5.2 · unverified

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