On the probability of n equidistant points in high-dimensional lattices

TL;DR

This paper analyzes the asymptotic probability that n points in high-dimensional lattices are equidistant, deriving explicit formulas using the multidimensional local limit theorem and lattice analysis.

Contribution

It provides a new asymptotic formula for the probability of equidistant points in high dimensions, extending to finitely supported distributions and different distance measures.

Findings

Probability asymptotically behaves as C_n/d^{(m-1)/2}

Explicit constant derived from first 4 moments of X

Method employs multidimensional local limit theorem

Abstract

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to \infty \quad \text{and} \quad m = \binom{n}{2}, \] with an explicit constant in terms of the first 4 moments of $X$. Moreover, we generalise this result to encompass all finitely supported $X$, as well as under different distances. Our method relies on the relatively rarely used multidimensional local limit theorem and an analysis of the lattice on $\mathbb{Z}^{\binom{n}{2}}$ spanned by the image of the \emph{overlapping} map \[ H : \{0,1\}^n \to \{0,1\}^{\binom{n}{2}}, \quad (v_1, \dots, v_n) \mapsto \Bigl( \mathbf{1}_{\{v_i \neq v_j\}} \Bigr)_{1 \le i < j \le n}. \]