# Bryce McLaughlin

Harvey Mudd College Mathematics 2017

Thesis Proposal: | An Incidence Approach to the Distinct Distances Problem |
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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(n^{1/2}) 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(n^{1-&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.