Browse > Article
http://dx.doi.org/10.4134/BKMS.b200937

ON ERDŐS CHAINS IN THE PLANE  

Passant, Jonathan (Department of Mathematics University of Rochester)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.5, 2021 , pp. 1279-1300 More about this Journal
Abstract
Let P be a finite point set in ℝ2 with the set of distance n-chains defined as ∆n(P) = {(|p1 - p2|, |p2 - p3|, …, |pn - pn+1|) : pi ∈ P}. We show that for 2 ⩽ n = O|P|(1) we have ${\mid}{\Delta}_n(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^n}{{\log}^{\frac{13}{2}(n-1)}{\mid}P{\mid}}}$. Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in [28], which allows one to iterate efficiently on intersecting nested subsets of Guth-Katz lines. Let G is a simple connected graph on m = O(1) vertices with m ⩾ 2. Define the graph-distance set ∆G(P) as ∆G(P) = {(|pi - pj|){i,j}∈E(G) : pi, pj ∈ P}. Combining with results of Guth and Katz [17] and Rudnev [28] with the above, if G has a Hamiltonian path we have ${\mid}{\Delta}_G(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^{m-1}}{\text{polylog}{\mid}P{\mid}}}$.
Keywords
Incidence geometry; distinct distances;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Bennett, A. Iosevich, and K. Taylor, Finite chains inside thin subsets of ℝd, Anal. & PDE 9 (2016), no. 3, 597-614.   DOI
2 A. Iosevich and J. Passant, Finite point configurations in the plane, rigidity and Erdos problems, Proc. Steklov Inst. Math. 303 (2018), no. 1, 129-139.   DOI
3 B. Bukh and A. Hubard, Space crossing numbers, Combin. Probab. Comput. 21 (2012), no. 3, 358-373. https://doi.org/10.1017/S096354831100040X   DOI
4 N. Frankl and A. Kupavskii, Almost sharp bounds on the number of discrete chains in the plane, in 36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl-Leibniz-Zentrum fur Informatik, 2020.
5 Y. Ou and K. Taylor, Finite point configurations and the regular value theorem in a fractal setting, arXiv preprint arXiv:2005.12233, 2020.
6 M. Sharir and N. Solomon, Incidences between points and lines on two-and three-dimensional varieties, Discrete & Computational Geometry 59 (2018), no. 1, 88-130.   DOI
7 V. Chan, I. Laba, and M. Pramanik, Finite configurations in sparse sets, J. Anal. Math. 128 (2016), 289-335. https://doi.org/10.1007/s11854-016-0010-3   DOI
8 J. Bourgain, A Szemeredi type theorem for sets of positive density in Rk, Israel J. Math. 54 (1986), no. 3, 307-316. https://doi.org/10.1007/BF02764959   DOI
9 P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Exposition. Math. 17 (1999), no. 4, 289-311.
10 E. A. Palsson, S. Senger, and A. Sheffer, On the number of discrete chains, arXiv preprint arXiv:1902.08259, 2019.
11 G. Tardos, On distinct sums and distinct distances, Adv. Math. 180 (2003), no. 1, 275-289. https://doi.org/10.1016/S0001-8708(03)00004-5   DOI
12 A. Iosevich and H. Parshall, Embedding distance graphs in finite field vector spaces, J. Korean Math. Soc. 56 (2019), no. 6, 1515-1528. https://doi.org/10.4134/JKMS.j180776   DOI
13 L. Guth, A. Iosevich, Y. Ou, and H. Wang, On Falconer's distance set problem in the plane, Invent. Math. 219 (2020), no. 3, 779-830. https://doi.org/10.1007/s00222-019-00917-x   DOI
14 L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane, Ann. of Math. (2) 181 (2015), no. 1, 155-190. https://doi.org/10.4007/annals.2015.181.1.2   DOI
15 A. Iosevich, G. Jardine, and B. McDonald, Cycles of arbitrary length in distance graphs on 𝔽q, arXiv preprint arXiv:2101.00748, 2021.
16 E. Landau, Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, 1909.
17 M. Rudnev, On the number of classes of triangles determined by n points in ℝ2, arXiv preprint arXiv:1205.4865, 2012.
18 T. Ziegler, Nilfactors of ℝm-actions and configurations in sets of positive upper density in ℝm, J. Anal. Math. 99 (2006), 249-266. https://doi.org/10.1007/BF02789447   DOI
19 N. Lyall and A. Magyar, Distance graphs and sets of positive upper density in ℝd, Anal. & PDE 13 (2020), no. 3, 685-700.   DOI
20 L. Moser, On the different distances determined by n points, Amer. Math. Monthly 59 (1952), 85-91. https://doi.org/10.2307/2307105   DOI
21 M. Rudnev, Note on the number of hinges defined by a point set in ℝ2, Combinatorica 40 (2020), no. 5, 749-757. https://doi.org/10.1007/s00493-020-4171-4   DOI
22 J. Solymosi and G. Tardos, On the number of k-rich transformations, in Computational geometry (SCG'07), 227-231, ACM, New York. https://doi.org/10.1145/1247069.1247111
23 J. Solymosi and C. D. Toth, Distinct distances in the plane, Discrete & Computational Geometry 25 (2001), no. 4, 629-634.   DOI
24 F. R. K. Chung, The number of different distances determined by n points in the plane, J. Combin. Theory Ser. A 36 (1984), no. 3, 342-354. https://doi.org/10.1016/0097-3165(84)90041-4   DOI
25 F. R. K. Chung, E. Szemeredi, and W. T. Trotter, The number of different distances determined by a set of points in the euclidean plane, Discrete & Computational Geometry 7 (1992), no. 1, 1-11.   DOI
26 F. de Zeeuw, A short proof of rudnev's point-plane incidence bound, arXiv preprint arXiv:1612.02719, 2016.
27 G. Elekes and M. Sharir, Incidences in three dimensions and distinct distances in the plane, Combin. Probab. Comput. 20 (2011), no. 4, 571-608. https://doi.org/10.1017/S0963548311000137   DOI
28 P. Erdos, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248-250. https://doi.org/10.2307/2305092   DOI
29 H. Furstenberg, Y. Katznelson, and B. Weiss, Ergodic theory and configurations in sets of positive density, in Mathematics of Ramsey theory, 184-198, Algorithms Combin., 5, Springer, Berlin, 1990. https://doi.org/10.1007/978-3-642-72905-8_13
30 K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), no. 2, 206-212 (1986). https://doi.org/10.1112/S0025579300010998   DOI
31 J. Garibaldi, A. Iosevich, and S. Senger, The Erdos distance problem, Student Mathematical Library, 56, American Mathematical Society, Providence, RI, 2011. https://doi.org/10.1090/stml/056
32 L. Guth, Polynomial methods in combinatorics, University Lecture Series, 64, American Mathematical Society, Providence, RI, 2016. https://doi.org/10.1090/ulect/064
33 K. J. Falconer and J. M. Marstrand, Plane sets with positive density at infinity contain all large distances, Bull. London Math. Soc. 18 (1986), no. 5, 471-474. https://doi.org/10.1112/blms/18.5.471   DOI