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Almost sharp bounds on the number of discrete chains in the plane

Frankl, Nóra and Kupavskii, Andrey (2020) Almost sharp bounds on the number of discrete chains in the plane. In: Cabello, Sergio and Chen, Danny Z., (eds.) 36th International Symposium on Computational Geometry, SoCG 2020. Leibniz International Proceedings in Informatics, LIPIcs. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. ISBN 9783959771436

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Identification Number: 10.4230/LIPIcs.SoCG.2020.48


The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ = (δ1,..., δk) of k distances, a (k + 1)-tuple (p1,..., pk+1) of distinct points in Rd is called a (k, δ)-chain if kpj − pj+1k = δj for every 1 ≤ j ≤ k. What is the maximum number Ckd(n) of (k, δ)-chains in a set of n points in Rd, 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.

Item Type: Book Section
Additional Information: © 2022 The Author(s).
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 18 Aug 2022 14:18
Last Modified: 18 Aug 2022 23:04

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