Abstract: Let \( X=\{x_1,\dots,x_n\}\subset \mathbb{R}^2 \) be a finite set of points of diameter at most \(1\). It is natural to expect that if many pairs \( (x_i,x_j) \) lie at distance close to \(1\) from each other, then some clustering phenomenon must occur, implying that a significant number of these pairs are also very close to each other.
We prove that there exists a universal constant \( c>0 \) such that for all \( 0<\varepsilon<1 \), whenever \( n \) is large enough, we have:
$$ \big|\{(i,j):\|x_i-x_j\|\le \varepsilon\}\big| \ge c\cdot \varepsilon^{1/2}\cdot \big|\{(i,j):\|x_i-x_j\|\ge 1-\varepsilon\}\big|. $$This confirms a recent conjecture of Steinerberger, who asked whether the \( \varepsilon^{1/2} \) ratio is the best possible. We also study a two-parameter version of Steinerberger's question by considering the number of pairs at distance \( <\varepsilon_1 \) and at distance \( >1-\varepsilon_2 \).
We show that in this case the optimal ratio is \( \varepsilon_1^2\cdot\varepsilon_2^{-3/2} \). The proof proceeds by introducing an auxiliary graph associated with the set \( X \) and reducing the problem to bounding the largest eigenvalue of its adjacency matrix.
Our main result is the product of human-AI interactions using ChatGPT 5.4.