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3 results

  • AN INCIDENCE RESULT FOR WELL-SPACED ATOMS IN ALL DIMENSIONS

  • Part of
  • PETER J. BRADSHAW
  • Journal:
    Journal of the Australian Mathematical Society / Volume 115 / Issue 1 / August 2023
    Published online by Cambridge University Press:
    20 February 2023, pp. 58-72
    Print publication:
    August 2023
    • Article
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  • We prove an incidence result counting the k-rich $\delta $-tubes induced by a well-spaced set of $\delta $-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi–Trotter theorem and holds in all dimensions $d \geq 2$. We also prove an analogue of Beck’s theorem for $\delta $-atoms and $\delta $-tubes as an application of our result.

  • 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.

Polynomial Methods and Incidence Theory
  • Adam Sheffer
  • Published online:
    17 March 2022
    Print publication:
    24 March 2022
  • The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erdős's distinct distances problem, the joints problem, as well as others, thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics, discrete geometry, and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications, with a focus on incidence theory. Based on the author's own teaching experience, the text requires a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail, with many examples, warm-up proofs, and exercises included. An appendix provides a quick reminder of basic results and ideas.