  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.
