A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions

Abstract

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having density at least s, and having the least number of 3-term arithemtic progressions among all sets of density at least s, then S is nearly translation invariant in a very strong sense. Namely, there exists 0 <= b <= q-1 such that |S intersect (S + bj)| = (1-g(s))|S|, for every 0 < j < q^d, where g(s) -> 0 as s -> 0. A curious feature of the proof is that Behrend's construction on large subsets of {1,2,...,x} containing no 3-term a.p., is a key ingredient.