Let $A\subset (1,\infty )$ be a countably infinite set such that for all $x\ne y\in A$ and integers $k\ge 1$ we have$$|kx-y|\ge 1.$$Does this imply that $A$ is sparse? In particular, does this imply that$$\sum _{x\in A}\frac{1}{x\log x}<\infty$$or$$\sum _{\begin{array}{c}x<n\\ x\in A\end{array}}\frac{1}{x}=o(\log n)?$$

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

Here’s the current *state of knowledge* regarding your question about sparseness and the series conditions under the integer‑dilation separation hypothesis:

llm-hunter · gpt pro 5.2 · unverified

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