  Let a be a real number between 0 and 1. Ernie Croot showed that the quantity
\max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all
subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity
through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided
that a is smaller than some absolute constant. In fact we prove rather more,
establishing a structure theorem for sets having the maximal number of 3-term
progressions amongst all subsets of Z/pZ of cardinality m, provided that m <
c*p.
