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Crescent configurations in normed spaces

Fish, Sara and King, Dylan and Miller, Steven J. and Palsson, Eyvindur A. and Wahlenmayer, Catherine (2020) Crescent configurations in normed spaces. Integers, 20 . Art. No. A96. ISSN 1553-1732.

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We study the problem of crescent configurations, posed by Erdos in 1989. A crescent configuration is a set of n points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each 1 ≤ i ≤ n - 1, there exists a distance which occurs exactly i times. Constructions of sizes n ≤ 8 have been provided by Liu, Palásti, and Pomerance. Erdos conjectured that there exists some N for which there do not exist crescent configurations of size n for all n ≥ N. We extend the problem of crescent configurations to general normed spaces (ℝ², II · II) by studying strong crescent configurations in II · II. In an arbitrary norm II · II, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations of size n ≤ 6 in the Euclidean norm and of size n ≤ 8 in the taxicab and Chebyshev norms. When defining strong crescent configurations, we introduce the notion of line-like configurations in II · II. A line-like configuration in II · II is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common II · II circle. We prove that for n ≥ 7, every line-like crescent configuration of size n in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.

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URLURL TypeDescription Paper
Miller, Steven J.0000-0003-4904-4605
Additional Information:© 2020 The Author(s), under a Creative Commons Attribution 4.0 International License. Received: 11/18/19 , Accepted: 10/27/20, Published: 11/2/20. The authors were partially supported by NSF grant DMS1659037. The second author was partially supported by the N.S. Reynolds Scholarship Committee at Wake Forest University, the third author was supported by NSF grant DMS1561945 and the fourth author was supported by Simons Foundation Grant #360560. We also thank Charles Devlin IV for helpful conversations about this problem.
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Wake Forest UniversityUNSPECIFIED
Simons Foundation360560
Record Number:CaltechAUTHORS:20210329-133805377
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108572
Deposited By: Tony Diaz
Deposited On:30 Mar 2021 23:27
Last Modified:30 Mar 2021 23:27

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