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.

We started our study of the distinct distances problem in Section 1.6. The mathematicians Elekes and Sharir used to discuss this problem. Around the turn of the millennium, Elekes discovered a reduction from this problem to a problem about intersections of helices in R^3. Elekes said that, if something happens to him, then Sharir should publish their ideas.

Elekes passed away in 2008 and, as requested, Sharir then published their ideas. Before publishing, Sharir simplified the reduction so that it led to a problem about intersections of parabolas in R^3. Sharing the reduction with the general community had surprising consequences. Hardly any time had passed before Guth and Katz managed to apply the reduction to almost completely solve the distinct distances problem.

In this chapter we study the reduction of Elekes, Sharir, Guth, and Katz. This reduction is based on parameterizing rotations of the plane as points in R^3. As a warmup, we begin with a problem about distinct distances between two lines.

After the long and technical proof of the distinct distances theorem, we move to a lighter chapter. In this chapter we study two additional distinct distances problems. We first show that every planar point set contains a large subset that does not span any distance more than once. We then study the structural distinct distances problem: characterizing the point sets that span a small number of distinct distances.

We also study a problem that does not involve distinct distances, but relies on a variant of Theorem 9.2. This problem considers sets of intervals in the plane that span many trapezoids.

In Chapter 7 we studied the ESGK framework. This was a reduction from the distinct distances problem to a problem about pairs of intersecting lines in R^3. In the current chapter we further reduce the problem to bounding the number of rich points of lines in R^3. We solve this incidence problem with a more involved variant of the constant-degree polynomial partitioning technique. This completes the proof of the Guth–Katz distinct distances theorem.

The original proof of Guth and Katz is quite involved. We study a simpler proof for a slightly weaker variant of the distinct distances theorem. This simpler proof was introduced by Guth and avoids the use of tools such as flat points and properties of ruled surfaces.

In Chapter 11 we derived general point-variety incidence bounds in R^d. We now study two applications of these bounds. These applications do not require reading any part of Chapter 11, except for the statement of Theorem 11.3. The first application comes from discrete geometry and is another distinct distances problem. Specifically, it is a distinct distances problem with local properties. The second application is a discrete Fourier restriction problem from harmonic analysis. For simplicity, we only discuss the combinatorial aspect of that problem. This aspect is the additive energy of points on a hypersphere.