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 $$C_n\cdot \frac{1}{d^{(m-1)\mathrm{/}2}}\text{for}d\to \mathrm{\infty }\text{and}m=\left(\genfrac{}{}{0px}{}{n}{2}\right),$$ 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 ${\mathbb{Z}}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}$ spanned by the image of the \emph{overlapping} map $$H:\{0,1{\}}^n\to \{0,1{\}}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)},(v_1,\dots ,v_n)\mapsto ({\mathbf{1}}_{\{v_i\mathrm{\ne }{v}_j\}}{)}_{1\le i<j\le n}\mathrm{.}$$