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Published online by Cambridge University Press: 17 March 2022
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.
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