Abstract
We solve two related extremal-geometric questions in the n-dimensional space \(\mathbb{R}_{\infty }^{n}\) equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in \(\mathbb{R}_{\infty }^{n}\) equals \({{2}^{{n + 1}}} - 1\). A sequence is right-equidistant if each of the points is at the same distance from all the succeeding points. Second, we prove that the maximum number of points in \(\mathbb{R}_{\infty }^{n}\) with pairwise odd distances equals 2n. We also obtain partial results for both questions in the n-dimensional space \(\mathbb{R}_{1}^{n}\) with the Manhattan distance.
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Notes
This example is not unique. Moreover, there exist continually many pairwise nonisometric sequences of maximum size.
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ACKNOWLEDGMENTS
We are grateful to Ilya Bogdanov for the simplification of the original proof of Theorem 2.
Funding
This work was supported by the Russian Science Foundation, grant no. 22-21-00368.
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Translated by I. Ruzanova
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Golovanov, A.I., Kupavskii, A.B. & Sagdeev, A.A. Odd-Distance Sets and Right-Equidistant Sequences in the Maximum and Manhattan Metrics. Dokl. Math. 106, 340–342 (2022). https://doi.org/10.1134/S106456242205012X
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DOI: https://doi.org/10.1134/S106456242205012X