A Structural Theorem for Sets with Few Triangles

We show that if a finite point set \(P\subseteq {\mathbb {R}}^2\) has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds.

• There is a \(\sigma >0\) and a line l which contains \(\Omega (|P|^\sigma )\) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing \(\Omega (|P|^\sigma )\) points of P.

• There is a circle \(\gamma \) which contains a positive proportion of P.

This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.