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Source. Karamanlis, published p. 6, Proposition 8 (canonical PDF). The corresponding arXiv v1–v3 result is Proposition 3.6.

Statement. For every integer $k\ge1$, every finite $X\subseteq\mathbb R^k$ and every $\delta>0$, there are an integer $m\ge2$ and a real $r>0$ such that $X$ has a $\delta$-embedding into $T_{m,r}^{k}$.

Proof. For each coordinate projection $\pi_a(X)$ containing at least two points, list its positive pairwise distances. There are finitely many such distances over all coordinates. Choose one integer $n_0\ge1$ large enough that every listed distance belongs to $[1/n_0,n_0]$. If there are no listed distances, choose $n_0=1$.

Apply the repaired Lemma 7 with tolerance $\delta/k$, using a single integer

$$n\ge\max\{2,n_0,2\pi n_0^3k/\delta\}.$$

It gives the same $m=n^3$ and $r=n_0n/(2\pi)$ for every nonsingleton projection. For a singleton projection, map its sole value to one vertex of the same polygon; its distance error is zero and the map is injective. An empty $X$ has the empty embedding, so assume $X$ nonempty below. Let $f_a:\pi_a(X)\to T_{m,r}$ denote the chosen coordinate maps.

Set $f(x)=(f_1(\pi_1x),\ldots,f_k(\pi_kx))$. Distinct points of $X$ differ in some coordinate, where $f_a$ is injective, so $f$ is injective. Orthogonality and the triangle inequality give

$$\begin{aligned} \left|\|f(x)-f(x')\|^2-\|x-x'\|^2\right| &\le\sum_{a=1}^{k} \left|\|f_a(\pi_ax)-f_a(\pi_ax')\|^2-|\pi_ax-\pi_ax'|^2\right|\\ &<k(\delta/k)=\delta. \end{aligned}$$

This proves the assertion. A configuration initially in $\mathbb R^0$ is empty or a singleton and may first be placed in $\mathbb R$. $\square$

Source precision. The common separation bound and singleton-coordinate cases expand the source's choice of a common $(m,r)$. This is an existential approximation; no equality of distances is claimed here. The missing correction is supplied by Proposition 11.