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Discrete & Computational Geometry

, Volume 58, Issue 1, pp 232–253 | Cite as

Curves in \(\mathbb {R}^4\) and Two-Rich Points

  • Larry Guth
  • Joshua ZahlEmail author
Article

Abstract

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in \(\mathbb {R}^4\). Specifically, we show that an arrangement of n algebraic curves determines at most \(C_\varepsilon n^{4/3+3\varepsilon }\) two-rich points, provided at most \(n^{2/3-2\varepsilon }\) curves lie in any low degree hypersurface and at most \(n^{1/3-\varepsilon }\) curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.

Keywords

Incidence geometry Combinatorial geometry Polynomial partitioning 

Notes

Acknowledgments

Larry Guth research was supported by a Simons Investigator award. Joshua Zahl research was supported by a NSF Postdoctoral Fellowship.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of British ColumbiaVancouverCanada

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