An incidence approach to the discrete distances problem

04/10/2018 - 12:15pm
04/10/2018 - 1:10pm
Bryce McLaughlin (HMC)

In 1946, Erdős 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. Erdős showed that any point set must realize at least Ω(n^{1/2}) distances, but could only provide a construction which offered Ω(n/√log(n)) distances. He conjectured that the actual minimum number of distances was Ω(n^{1-ε}) fo any ε>0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdős' 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 Ω(n/log(n)) distances. In this talk we will explore how incidence theory played a role in this process and expand upon recent work by Adam Sheffer and Cosmin Pohoata who use geometric incidences to get 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 Ω(n/√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.

Millikan 2099, Pomona College

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