Loose Embeddings in a Fixed Euclidean Space: Countable Counterexamples and Flat Tori

A loose embedding of a metric space \((X,d)\) in \(\mathbb{R}^N\), in the sense of Chirvasitu, is a continuous map \(\varphi\) such that \(|\varphi x\)\({}-\varphi y|\)\({}=|\varphi x'\)\({}-\varphi y'|\) if and only if \(d(x,y)=d(x',y')\). In his abstract in an Oberwolfach report of 2019, Chirvasitu asked whether a compact metric space all of whose finite subspaces are loosely embeddable in \(\mathbb{R}^N\), for one fixed \(N\), is itself loosely embeddable there, and whether every compact Riemannian manifold with its geodesic distance satisfies this hypothesis. We answer both questions negatively, the first in the reading in which the conclusion asks for a loose embedding in the same \(\mathbb{R}^N\), which is the reading of the problem record we work from. For every \(N\ge1\) we construct a countable compact metric space all of whose finite subspaces are loosely embeddable in \(\mathbb{R}^N\) but which is not loosely embeddable in \(\mathbb{R}^N\). This does not answer Chirvasitu's later Question 2.1 of arXiv:2101.01242, which asks only for a loose embedding in some finite-dimensional Euclidean space: our example for \(N=1\) is loosely embeddable in \(\mathbb{R}^2\), and that question remains open. For the second question we prove that every flat torus of dimension at least \(2\) has, for every \(N\), a finite subset that is not loosely embeddable in \(\mathbb{R}^N\). This proof is non-constructive: for general \(N\) it gives no explicit finite subset and no bound on its size (for \(N\le2\) we give an explicit four-point subset of the hexagonal torus). The argument also shows that no flat torus of dimension at least \(2\) is loosely embeddable in any Euclidean space, whereas round spheres and real and complex projective spaces are. The key lemma for tori is the flat-torus analogue of a theorem of Heitzig on maps between Euclidean spaces that preserve equality of distances, and its proof follows his method. The definitions and the questions are Chirvasitu's. We found no printed answer to either question in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.