Abstract
We generalize the Szemerédi–Trotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each \(i=0,1,\ldots ,d-1\), we are given a finite set \(S_i\) of i-flats in \({\mathbb {R}}^d\) or in \({\mathbb {C}}^d\), and a (complete) flag is a tuple \((f_0,f_1,\ldots ,f_{d-1})\), where \(f_i\in S_i\) for each i and \(f_i\subset f_{i+1}\) for each \(i=0,1,\ldots ,d-2\). Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in \({\mathbb {R}}^3\) such that among the lines incident to a point, at most O(1) of them can be coplanar, (ii) incidences with Legendrian lines in \({\mathbb {R}}^3\), a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in \({\mathbb {R}}^3\) (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.
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Notes
Clearly, the condition that each point is incident to at most b coplanar lines continues to hold in the inductive subproblems.
References
Baston, R.J., Eastwood, M.G.: The Penrose Transform: Its Interaction with Representation Theory. Clarendon Press, New York (1989)
Buczyński, J.: Legendrian subvarieties of projective spaces. Geom. Dedicata 118, 87–103 (2006)
deGosson, M.A.: Symplectic Geometry and Quantum Mechanics, vol. 166. Springer, NewYork (2006)
Elekes, G.: Sums versus products in number theory, algebra and Erdős geometry—a survey. In: Paul Erdős, and His Mathematics II. Bolyai Math. Soc., Stud. 11, pp. 241–290. Budapest (2002)
Elekes, G., Kaplan, H., Sharir, M.: On lines, joints, and incidences in three dimensions. J. Comb. Theory Ser. A 118, 962–977 (2011)
Guth, L., Katz, N.H.: Algebraic methods in discrete analogs of the Kakeya problem. Adv. Math. 225, 2828–2839 (2010)
Guth, L., Katz, N.H.: On the Erdős distinct distances problem in the plane. Ann. Math. 181, 155–190 (2015)
Sharir, M., Welzl, E.: Point-line incidences in space. Comb. Probab. Comput. 13, 203–220 (2004)
Szemerédi, E., Trotter Jr, W.T.: Extremal problems in discrete geometry. Combinatorica 3, 381–392 (1983)
Tóth, C.D.: The Szemerédi–Trotter theorem in the complex plane. Combinatorica 35, 95–126 (2015)
Zahl, J.: A Szemerédi–Trotter type theorem in \({\mathbb{R}}^4\). Discrete Comput. Geom. 54, 513–572 (2015)
Acknowledgments
Saarik Kalia and Ben Yang would like to thank Larry Guth for suggesting this project and providing tools which were crucial in the discovery of our results. They would also like to thank David Jerison and Pavel Etingof for repeatedly offering insightful input into our project and providing future directions of research. Finally, they would like to thank the SPUR program for allowing them the opportunity to conduct this research. Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.
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Kalia, S., Sharir, M., Solomon, N. et al. Generalizations of the Szemerédi–Trotter Theorem. Discrete Comput Geom 55, 571–593 (2016). https://doi.org/10.1007/s00454-016-9759-5
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DOI: https://doi.org/10.1007/s00454-016-9759-5