Structural Results For Planar Sets With Many Similar Subsets

Consider a k-element subset P of the plane. It is known that the maximum number of sets similar to P that can be found among n points in the plane is Θ(n ²) if and only if the cross ratio of any quadruplet of points in P is algebraic [3], [9].

In this paper we study the structure of the extremal n-sets A which have cn ² similar copies of P. As our main result we prove the existence of large lattice-like structures in such sets A. In particular we prove that, for n large enough, A must contain m points in a line forming an arithmetic progression, or m × m lattices, when P is not cocyclic or collinear. On the other hand we show that for cocyclic or collinear sets P, there are n-element sets A with c _P n ² copies of P and without k × k lattice subsets.