Bryce McLaughlin

Harvey Mudd College Mathematics 2017

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Thesis Proposal: An Incidence Approach to the Distinct Distances Problem
Thesis Advisor: Prof. Mohamed Omar
Second Reader: Prof. Dagan Karp
E-Mail: bmclaughlin@g.hmc.edu

An Incidence Approach to the Distinct Distances Problem

In 1946, Erdös posed the distinct distances problem, which asks for the minimum number of distinct distances that any set of n points in the real plane must realize. Erdös showed that any point set must realize at least &Omega(n1/2) distances, but could only provide a construction which offered &Omega(n/&radic(log(n)))$ distances. He conjectured that the actual minimum number of distances was &Omega(n1-&epsilon) for any &epsilon > 0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdös' conjecture seemed to hold until in 2010 Larry Guth and Nets Hawk Katz used an incidence theory approach to show any point set must realize at least &Omega(n/log(n)) distances. In this thesis we will explore how incidence theory played a roll in this process and expand upon recent work by Adam Sheffer and Cosmin Pohoata, using geometric incidences to achieve bounds on the bipartite variant of this problem. A consequence of our extensions on their work is that the theoretical upper bound on the original distinct distances problem of &Omega(n/&radic(log(n))) holds for any point set which is structured such that half of the n points lies on an algebraic curve of arbitrary degree.