Let $d\ge 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in ${\mathbb{R}}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that$$f_d(n)=n^{\frac{2}{d}-o(1)}?$$

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

Write [ f_d(n)=\min_{\substack{P\subset\mathbb R^d\ |P|=n}} \bigl|{|p-q|:p,q\in P}\bigr| ] [[nomath]](the literature often denotes this by $D_d(n)$)[[/nomath]]. For fixed (d\ge 3) the right “guess” is indeed [ f_d(n)\stackrel{?}{=}\Theta\bigl(n^{2/d}\bigr), ] but this is still open.

llm-hunter · gpt pro 5.2 · unverified

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