Appendix A
Now I shall turn to how well space approximates Euclidean space under the mixed account. Under this account, an atomistic space can be represented by a set of points with a shortest path metric that assigns some pairs of points real-valued distances (bounded by a finite number) and derives other distances as their least sums.
We will understand “approximation” in terms of “almost isometry.” Let e(p, q) be the Euclidean distance between two points p, q in Euclidean space. Let \(\epsilon , r\) be two positive numbers. A metric space X with a metric d is \(\epsilon \)-isometric to Euclidean space E with regard to r iff there is a map f from X to E such that (1) for \(x,y\in X\), we have
$$\begin{aligned} 1-\epsilon \le \frac{e(f(x),f(y))}{d(x, y)}\le 1+\epsilon \end{aligned}$$
(the smallest \(\epsilon \) such that f satisfies this condition is called the distortion of f);Footnote 27 (2) for every \(p\in E\), there is a \(x\in X\) such that \(e(p,f(x))\le r\). In other words, the embedded points cover E reasonably well so that there are no obvious “clusters” and “holes.”
Theorem A.1
For any \(\epsilon \) and r, there is a set of points with a shortest path metric (with distances being bounded by a finite number) that is \(\epsilon \)-isometric to Euclidean space with regard to r.
Proof
For brevity, I will resort to the following abbreviations when applicable. Given an embedding f of a metric space into Euclidean space, for any points x, y in the space, let \(\Vert xy\Vert _f=e(f(x),f(y))\) (the subscript “f” is omitted if it is clear which embedding we refer to). Also, for any points p, q in Euclidean space, let \(\Vert pq\Vert =e(p,q).\)
Let G be an embedding of an infinite set X to Euclidean space E such that there is an r such that for any \(p\in E\), we can find an \(x\in X\) with \(e(p,G(x))<r\). (For example, if G maps members of X to Euclidean points represented by pairs of integers, then r in question is at least \(\sqrt{2}/2\).) We will construct a metric over X such that the resulting metric space is \(\epsilon \)-isometric to Euclidean space under G, where \(\epsilon \) is a small number we choose.
M is a real-number parameter that will play an important role in assigning weights and in determining the distortion of the intended embedding. For any \(x,y\in X\), if \(\Vert xy\Vert > M\), we can find a sequence of points \(p_1,p_2,\ldots p_n\) in E such that \(p_0=G(x)\), \(p_n=G(y)\), \(\Vert p_0p_1\Vert =\Vert p_1p_2\Vert =\cdots =\Vert p_{n-2}p_{n-1}\Vert =M\) and \(\Vert p_{n-1}p_n\Vert <M\). Let \(N=\Vert p_{n-1}p_n\Vert \). Consider \(p_i,p_{i+1}\), where \(i=1,\ldots ,n-2\). We can find \(x_i,x_{i+1}\in X\) such that \(e(G(x_i),p_i)<r\) and \(e(G(x_{i+1}),p_{i+1})<r\). We know that the largest distance between points on two circles is equal to the distance between their centers plus their radii.Footnote 28 Thus, \(\Vert x_ix_{i+1}\Vert <M+2r\). Now, for any two \(a,b\in X\), if \(\Vert ab\Vert <M+2r\), then let them be connected by an edge with the weight \(d(a,b)=\Vert ab\Vert \); otherwise, a, b are not connected by an edge. Then, \(M\le d(x_i,x_{i+1})<M+2r\). Moreover, it’s easy to see that \(M\le d(x,x_1)\le M+r\) and \(N\le d(x_{n-1},y)\le N+r\). It follows that \(d(x,y)\le d(x,x_1)+d(x_1,x_2)+\cdots +d(x_{n-1},y)< n\cdot (M+2r)+(N+r)\). Furthermore, if \(x, x_1, \ldots x_n, y\) is a shortest path, then \(d(x,y)=d(x,x_1)+d(x_1,x_2)+\cdots +d(x_{n-1},y)=\Vert xx_1\Vert +\cdots +\Vert x_{n-1}y\Vert \ge \Vert xy\Vert \). Thus, we have:
$$\begin{aligned} 1\le \frac{d(x,y)}{\Vert xy\Vert }<\frac{n\cdot (M+2r)+(N+r)}{nM+N}=1+\frac{(2n+1)r}{nM+N} \end{aligned}$$
The distortion \(\displaystyle \delta =\frac{d(x,y)}{\Vert xy\Vert }-1<\frac{(2n+1)r}{nM+N}<\frac{(2n+1)r}{nM}<\frac{3r}{M}\). Then, for any small positive number \(\epsilon \), we can make \(\delta <\epsilon \) by letting \(M=3r/\epsilon \). (Note that if we are only concerned with distances that involve a large n, we only need M to be \(2r/\epsilon \).) This completes the case for any \(x,y\in X\) with \(\Vert xy\Vert > M\). If \(\Vert xy\Vert \le M\), then we have \(d(x,y)=\Vert xy\Vert \), in which case there is no distortion. Therefore, we have found a metric space, in which all distances are bounded by \(3r/\epsilon +2r\), that is \(\epsilon \)-isometric to Euclidean space at any scale. \(\square \)