A Structural Theorem for Sets with Few Triangles

We show that if a finite point set P ⊆ R² has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds.
• There is a σ > 0 and a line l which contains Ω(|P|^σ ) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing Ω(|P|^σ ) points of P.
• There is a circle γ which contains a positive proportion of P.
This provides evidence for two conjectures of Erd˝os. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.