Large sets in finite fields are sumsets

For a prime p, a subset S of Z_p is a sumset if S = A + A for some A ⊂ Z_p. Let f (p) denote the maximum integer so that every subset S ⊂ Z_p of size at least p - f (p) is a sumset. The question of determining or estimating f (p) was raised by Green. He showed that for all sufficiently large p, f (p) ≥ frac(1, 9) log₂ p and proved, with Gowers, that f (p) < c p^{2 / 3} log^{1 / 3} p for some absolute constant c. Here we improve these estimates, showing that there are two absolute positive constants c₁, c₂ so that for all sufficiently large p,c₁ frac(sqrt(p), sqrt(log p)) ≤ f (p) < c₂ frac(p^{2 / 3}, log^{1 / 3} p) . The proofs combine probabilistic arguments with spectral techniques.