On the number of discrete chains

Palsson, Eyvindur; Senger, Steven; Sheffer, Adam

doi:10.1090/proc/15603

On the number of discrete chains
HTML articles powered by AMS MathViewer

by Eyvindur Ari Palsson, Steven Senger and Adam Sheffer

Proc. Amer. Math. Soc. 149 (2021), 5347-5358

DOI: https://doi.org/10.1090/proc/15603

Published electronically: September 28, 2021

PDF | Request permission

Abstract:

We study a generalization of the Erdős unit distance problem to chains of $k$ distances. Given $\mathcal {P},$ a set of $n$ points, and a sequence of distances $(\delta _1,\ldots ,\delta _k)$, we study the maximum possible number of tuples of distinct points $(p_1,\ldots ,p_{k+1})\in \mathcal {P}^{k+1}$ satisfying $|p_jp_{j+1}|=\delta _j$ for every $1\le j \le k$. We study the problem in $\mathbb {R}^2$ and in $\mathbb {R}^3$, and derive upper and lower bounds for this family of problems.

References

Boris Aronov and Micha Sharir, Cutting circles into pseudo-segments and improved bounds for incidences, Discrete Comput. Geom. 28 (2002), no. 4, 475–490. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). MR 1949895, DOI 10.1007/s00454-001-0084-1
Michael Bennett, Alexander Iosevich, and Krystal Taylor, Finite chains inside thin subsets of $\Bbb {R}^d$, Anal. PDE 9 (2016), no. 3, 597–614. MR 3518531, DOI 10.2140/apde.2016.9.597
Peter Brass, William Moser, and János Pach, Research problems in discrete geometry, Springer, New York, 2005. MR 2163782
P. Erdös, On sets of distances of $n$ points, Amer. Math. Monthly 53 (1946), 248–250. MR 15796, DOI 10.2307/2305092
P. Erdős, On sets of distances of $n$ points in Euclidean space, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 165–169 (English, with Russian summary). MR 141007
N. Frankl and A. Kupavskii, Almost sharp bounds on the number of discrete chains in the plane, arXiv:1912.00224, 2019.
S. Gunter, E. Palsson, B. Rhodes, and S. Senger, Bounds on point configurations determined by distances and dot products, arXiv:2011.15055, 2019, to appear in Nathanson M. (eds) Combinatorial and Additive Number Theory IV, CANT 2019, CANT 2020 Springer.
Haim Kaplan, Jiří Matoušek, Zuzana Safernová, and Micha Sharir, Unit distances in three dimensions, Combin. Probab. Comput. 21 (2012), no. 4, 597–610. MR 2942731, DOI 10.1017/S0963548312000144
Hanfried Lenz, Zur Zerlegung von Punktmengen in solche kleineren Durchmessers, Arch. Math. (Basel) 6 (1955), 413–416 (German). MR 76366, DOI 10.1007/BF01900515
Jiří Matoušek, The number of unit distances is almost linear for most norms, Adv. Math. 226 (2011), no. 3, 2618–2628. MR 2739786, DOI 10.1016/j.aim.2010.09.009
Y. Ou and K. Taylor, Finite point configurations and the regular value theorem in a fractal setting, arXiv:2005.12233, 2020.
J. Passant, On Erdős chains in the plane, arXiv:2010.14210, 2020.
Misha Rudnev, Note on the number of hinges defined by a point set in $\Bbb {R}^2$, Combinatorica 40 (2020), no. 5, 749–757. MR 4181765, DOI 10.1007/s00493-020-4171-4
J. Spencer, E. Szemerédi, and W. Trotter Jr., Unit distances in the Euclidean plane, Graph theory and combinatorics (Cambridge, 1983) Academic Press, London, 1984, pp. 293–303. MR 777185
Endre Szemerédi and William T. Trotter Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3-4, 381–392. MR 729791, DOI 10.1007/BF02579194
Joshua Zahl, Breaking the $3/2$ barrier for unit distances in three dimensions, Int. Math. Res. Not. IMRN 20 (2019), 6235–6284. MR 4031237, DOI 10.1093/imrn/rnx336
Joshua Zahl, An improved bound on the number of point-surface incidences in three dimensions, Contrib. Discrete Math. 8 (2013), no. 1, 100–121. MR 3118901