Large sets in finite fields are sumsets

For a prime p, a subset S of Zp is a sumset if S=A+A for some A⊂Zp. Let f(p) denote the maximum integer so that every subset S⊂Zp 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, and proved, with Gowers, that f(p)<cp2/3log1/3p 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,