Sam Mansfield

A structural theorem for sets with few triangles

In 1946, Erdős conjectured that the minimum number of distinct distances determined by a set of N points is N(log N)^{-1/2}, the number achieved by an N^{1/2} by N^{1/2} square lattice. This was (almost) solved by Guth and Katz in 2015, but a harder variant -- that any point set with only this number of distinct distances must have a lattice structure -- is wide open, and there are remarkably few results about the structure of such a set at all. Instead of considering single distances, we instead consider sets with few congruent triangles, showing such sets either contain a polynomially-rich line or a positive proportion of the set lies on a circle. Our methods include classical tools from additive combinatorics combined with geometric structure within the affine group. This is based on joint work with Jonathan Passant.