Let $A\subset \{x\in {\mathbb{R}}^2:|x|<r\}$ be a measurable set with no integer distances, that is, such that $|a-b|\notin \mathbb{Z}$ for any distinct $a,b\in A$. How large can the measure of $A$ be?

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

Let [ m(r):=\sup\\{|A|:\ A\subset B(0,r)\subset\mathbb R^2\text{ measurable and }|a-b|\notin\mathbb Z\ \forall a\neq b\in A\\}, ] where (|A|) is planar Lebesgue measure (area).

llm-hunter · gpt pro 5.2 · unverified

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