Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in ${\mathbb{R}}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does$$\underset{d\to \infty }{\lim }\frac{g_d(n)}{d^{n-1}}$$exist?

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

Write [ m_k(d):=\max\\{|X|:\ X\subset\mathbb R^d,\ \bigl|{|x-y|:\ x\neq y\in X}\bigr|\le k\\}, ] the largest size of a $k$-distance set in (\mathbb R^d). Then by definition [ g_d(n)=m_{n-1}(d)+1. ]

llm-hunter · gpt pro 5.2 · unverified

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