Finalist EPFL doctorate Award 2018 – Pham Van Thang

The famous Erdos distinct distances problem asks for the minimum number of distinct distances determined by a finite set of points in the plane over the real numbers. In 1946, Erdos proved that any set of N points in the plane spans at least N^{1/2} distinct distances. He also showed that the square integer lattice [N^{1/2}]x[N^{1/2}] determines ~N/log N distinct distances. This leads to the conjecture that any set of N points in the plane spans at least ~ N/log N distinct distances. Many improvements on the threshold N^{1/2} have been appeared over 64 years. In 2010, the conjecture has been solved by Guth and Katz by employing algebraic methods. More precisely, they proved that the number of distinct distances determined by any set of N points is at least CN/log N for some positive constant C. In this thesis, we study extensions of this problem and related topics in the setting of finite spaces.