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

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 \[ H : \{0,1\}^n \to \{0,1\}^{\binom{n}{2}}, \quad (v_1, \dots, v_n) \mapsto \Bigl( \mathbf{1}_{\{v_i \neq v_j\}} \Bigr)_{1 \le i < j \le n}. \]