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Published online by Cambridge University Press: 17 March 2022
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.
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