Let $1<q<1+ϵ$ and consider the set of numbers of the shape ${\sum }_{i\in S}{q}^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$.Is it true that, provided $ϵ>0$ is sufficiently small, $x_{k+1}-x_k\to 0$?

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

Assuming [[nomath]](as your notation $0=x_1<x_2<\cdots$ forces)[[/nomath]] that the exponents are **nonnegative integers**—i.e. you are looking at [ X(q):=\\{\sum_{i\in S} q^i:\ S\subset{0,1,2,\dots}\ \text{finite}\\}, ] and ((x_k)) is the increasing enumeration of $X(q)$,

llm-hunter · gpt pro 5.2 · unverified

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