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Almost Sharp Bounds on the Number of Discrete Chains in the Plane

Authors Nóra Frankl, Andrey Kupavskii

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  • Mathematics of computing → Combinatoric problems
Keywords
  • unit distance problem
  • unit distance graphs
  • discrete chains

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Abstract

The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ=(δ₁,… ,δ_k) of k distances, a (k+1)-tuple (p₁,… ,p_{k+1}) of distinct points in ℝ^d is called a (k,δ)-chain if ‖p_j-p_{j+1}‖ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3.

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Nóra Frankl and Andrey Kupavskii. Almost Sharp Bounds on the Number of Discrete Chains in the Plane. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.48

Author Details

Nóra Frankl
  • Department of Mathematics, London School of Economics and Political Science, UK
  • Laboratory of Combinatorial and Geometric Structures at MIPT, Moscow, Russia
Andrey Kupavskii
  • Moscow Institute of Physics and Technology, Moscow, Russia
  • Institute for Advanced Study, Princeton, NJ, US
  • G-SCOP, CNRS, Grenoble, France

Funding

The authors acknowledge the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no. 075-15-2019-1926.
  • Frankl, Nóra: Research was partially supported by the National Research, Development, and Innovation Office, NKFIH Grant K119670.
  • Kupavskii, Andrey: Research supported by the Russian Foundation for Basic Research (grant no. 18-01-00355) and the Council for the Support of Leading Scientific Schools of the President of the Russian Federation (grant no. HW-6760.2018.1). Research was directly supported by the IAS Fund for Math, the Director’s Fund and indirectly supported by the National Science Foundation Grant No. CCF-1900460. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Acknowledgements

We thank Konrad Swanepoel and the referees for helpful comments on the manuscript.

References

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