Long arithmetic progressions in critical sets

Given a prime p, we say that S ⊆ Z/pZ is a critical set of density d if and only if |S| = dp and S has the least number of arithmetic progressions among all other sets S having at least dp elements. In this context, an arithemtic progression is a triple of residue classes n, n+m,n+2m modulo p. Note that this includes “trivial” progressions, which are ones wherem ≡ 0 (mod p), as well as “non-trivial” progressions, which are ones where m 6≡ 0 (mod p). We also distinguish two different progressions, according to how they are ordered: The progression n, n +m,n + 2m is considered different from n+ 2m,n+m,n. The main result of the paper is the following theorem, which basically says that critical sets of positive density must have long arithemtic progressions.