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Let α ∈ [0, 1] be a real number. Ernie Croot ( Canad. Math. Bull.  51 (2008) 47–56) showed that the quantity max A  # (3‐term arithmetic progressions in  A )/p 2 , where  A  ranges over all subsets of ℤ/ p ℤ of size at most α  p , tends to a limit as  p  → ∞ through primes. Writing  c (α) for this limit, we show that  c (α)=α 2 /2 provided that α 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/ p  ℤ of cardinality  m , provided that  m  <  cp .

On the maximal number of 3‐term arithmetic progressions in subsets of ℤ/ p ℤ