Abstract
Let \(\mathcal {M}\) be a finite non-collinear set of points in the Euclidean plane, with the squared distance between each pair of points integral. Considering the points as lying in the complex plane, there is at most one positive square-free integer D, called the “characteristic” of \(\mathcal {M}\), such that a congruent copy of \(\mathcal {M}\) embeds in \(\mathbb {Q}(\sqrt{-D})\). We generalize the work of Yiu and Fricke on embedding point sets in \(\mathbb {Z}^2\) by providing conditions that characterize when \(\mathcal {M}\) embeds in the lattice corresponding to \(\mathcal {O}_{-D}\), the ring of integers in \(\mathbb {Q}(\sqrt{-D})\). In particular, we show that if the square of every ideal in \(\mathcal {O}_{-D}\) is principal and the distance between at least one pair of points in \(\mathcal {M}\) is integral, then \(\mathcal {M}\) embeds in \(\mathcal {O}_{-D}\). Moreover, if \(\mathcal {M}\) is primitive, so that the squared distances between pairs of points are relatively prime, and \(\mathcal {O}_{-D}\) is a principal ideal domain, then \(\mathcal {M}\) embeds in \(\mathcal {O}_{-D}\).
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Acknowledgments
We thank Sascha Kurz for introducing us to this problem in the context of his work on integer distance problems, and for confirming Conjecture 1 numerically. We thank several anonymous referees for their helpful corrections and suggestions to improve the exposition. We also thank Kiran Kedlaya for pointing out a connection between the 65 characteristics listed in Sect. 5.1 and Euler’s idoneal numbers (see Sect. 2.6 of [7]). Supported by NSF Grant DMS-1063070.
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Knopf, M., Milzman, J., Smith, D. et al. Lattice Embeddings of Planar Point Sets. Discrete Comput Geom 56, 693–710 (2016). https://doi.org/10.1007/s00454-016-9812-4
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DOI: https://doi.org/10.1007/s00454-016-9812-4
Keywords
- Lattice embedding
- Heronian triangle
- n-Cluster
- Ring of integers
- Maximal order
- Imaginary quadratic extension