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1 results

  • 1 - Incidences and Classical Discrete Geometry

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 1-24

  • Summary

    We begin our study of geometric incidences by surveying the field and deriving a few first bounds. In this chapter we only discuss classical discrete geometry, from before the discovery of the new polynomial methods. This makes the current chapter rather different than the rest of the book (outrageously, it even includes some graph theory). We also learn basic tricks that are used throughout the book, such as double counting, applying the Cauchy–Schwarz inequality, and dyadic decomposition.

    Topics that are discussed in this chapter: the Szemerédi–Trotter theorem, a proof of this theorem that relies on the crossing lemma, the unit distances problem, the distinct distances problem, a problem about unit area triangles, the sum-product problem, rich point, point-line duality.