On the Structure of Sets with Few Three-Term Arithmetic Progressions

Fix a prime p ≥ 3, and a real number 0 < α ≤ 1. Let S ⊂ Fpn be any set with the least number of solutions to x + y = 2z (note that this means that x, z, y is an arithmetic progression), subject to the constraint that |S| ≥ αp. What can one say about the structure of such sets S? In this paper we show that they are “essentially” the union of a small number… CONTINUE READING